Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 154637 9 pageshttpdxdoiorg1011552013154637

Research ArticleUniform Bounds of Aliasing and Truncated Errors inSampling Series of Functions from Anisotropic Besov Class

Peixin Ye and Yongjie Han

School of Mathematics and LPMC Nankai University Tianjin 300071 China

Correspondence should be addressed to Peixin Ye yepxnankaieducn

Received 1 May 2013 Accepted 11 June 2013

Academic Editor Yiming Ying

Copyright copy 2013 P Ye and Y Han This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Errors appear when the Shannon sampling series is applied to approximate a signal in real life This is because a signal may not bebandlimited the sampling series may have to be truncated and the sampled values may not be exact and may have to be quantizedIn this paper we truncate the multidimensional Shannon sampling series via localized sampling and obtain the uniform bounds ofaliasing and truncation errors for functions from anisotropic Besov class without any decay assumption The bounds are optimalup to a logarithmic factor Moreover we derive the corresponding results for the case that the sampled values are given by a linearfunctional and its integer translations Finally we give some applications

1 Introduction

Since Shannon introduced the sampling series in the land-mark paper [1] the Shannon sampling theorem has beena fundamental result in the field of information theory inparticular telecommunications and signal processing see [2ndash7] and the references therein The theorem states that abandlimited signal can be exactly recovered from an infinitesequence of its samples if the bandlimit is no greater thanhalf the sampling rate The theorem also leads to a formulafor reconstruction of the original function from its samplesWhen the function is not bandlimit the reconstructionexhibits imperfections known as aliasing Moreover in prac-tice the signal and the sampled values are not the accuratefunctional values So several types of errors such as aliasingerrors truncated errors jitter errors and amplitude errorsappear when the Shannon sampling series is applied toapproximate a signal in real life These types of errors havebeen widely studied under the assumption that signals satisfysome decay conditions at infinity see [8ndash13] On the otherhand one can avoid assumptions upon the decay rate ofthe initial signals by using localized sampling see [14ndash19]Recently the uniform bounds for truncated Shannon seriesbased on local sampling are derived for nonbandlimitedfunctions from Sobolev classes without decay assumption

see [18 19] In this paper we study errors in truncatedmultivariable Shannon sampling series via localized samplingby considering nonbandlimited functions from anisotropicBesov classes

It is well known that the sampling theorem is usuallyformulated for functions of a single variable Consequentlythe theorem is directly applicable to time-dependent signalsHowever the sampling theorem can be extended in a straight-forward way to functions of arbitrarily many variables Themultivariable sampling theorem can be used in the recon-struction of some types of images such as gray-scale images

We begin our discussion with the definitions of somefunction spaces Let 119871119901(R

119889) 1 le 119901 le infin be the space of

all 119901th power Lebesgue integrable functions onR119889 equippedwith the usual norm

10038171003817100381710038171198911003817100381710038171003817119871119901

= (intR119889

|119891(t)|119901119889t)1119901

(1)

for 1 le 119901 lt infin and1003817100381710038171003817119891

1003817100381710038171003817119871infin= ess sup

tisinR119889

1003816100381610038161003816119891 (t)1003816100381610038161003816 (2)

Set 119885119889 = 1 2 119889 For any vector k = (V119895 119895 isin 119885119889)

with positive coordinates we say an entire function ℎ is of

2 Abstract and Applied Analysis

exponential type k provided that for every 120576 gt 0 there exists apositive number 119888 such that for all complex vectors z = (119911119895

119895 isin 119885119889) isin C119889 we have the bound

|ℎ (z)| le 119888 exp( sum

119895isin119885119889

(V119895 + 120576)10038161003816100381610038161003816119911119895

10038161003816100381610038161003816) (3)

Denote by 119864k(C119889) the space of all entire functions of expo-

nential type k Let 119861k(R119889) be the subset of 119864k(C

119889) which are

bounded on R119889 Set

119861119901

k (R119889) = 119861k (R

119889) cap 119871119901 (R

119889) 1 le 119901 le infin (4)

Every vector k = (V119895 119895 isin 119885119889) isin R119889

+determines the rectangle

119868119889

k = prod

119895isin119885119889

[minusV119895 V119895] (5)

According to the Schwartz theorem [20]

119861119901

k (R119889) = 119891 isin 119871119901 (R

119889) supp 119891 sube 119868

119889

k (6)

where 119891 is the Fourier transform of 119891 in the sense ofdistribution For the case 119901 = 2 it is the classical Paley-Wiener theorem

Now we define anisotropic Besov space Suppose that 119896 isin

119873 and t isin R119889 For 119891 isin 119871119901(R119889) we define the 119896th partial

difference of 119891 in the 119897th coordinate direction e119897 at the pointt isin R119889 with step 119909119897 isin R by the formula

Δ119896

119909119897119891 (t) =

119896

sum

119894=0

(minus1)119894+119896

(119896

119894)119891 (t + 119894119909119897e119897) (7)

Let l = (1198971 119897119889) isin N119889 r = (1199031 119903119889) isin R119889

+ and 119897119894 gt 119903119894 for

119894 isin 119885119889 1 le 119901 and 120579 le infinWe say119891 isin 119861r119901120579(R119889

) if119891 isin 119871119901(R119889)

and the following seminorm is finite

10038171003817100381710038171198911003817100381710038171003817119887119903119894

119901120579

=

intR

(

10038171003817100381710038171003817Δ11989711989411990911989411989110038171003817100381710038171003817119871119901

|119909119894|119903119894+(1120579)

)

120579

119889119909119894

1120579

1 le 120579 lt infin

sup|119909119894| = 0

10038171003817100381710038171003817Δ11989711989411990911989411989110038171003817100381710038171003817119871119901

|119909119894|119903119894

120579 = infin

119894 = 1 119889

(8)

The linear space 119861r119901120579(R119889

) is a Banach space with the norm

10038171003817100381710038171198911003817100381710038171003817119861r119901120579

=1003817100381710038171003817119891

1003817100381710038171003817119871119901+ sum

119894isin119885119889

10038171003817100381710038171198911003817100381710038171003817119887119903119894

119901120579 (9)

and is called an anisotropic Besov space We introduce thenotation 119892(r) = (sum

119889

119894=1(1119903119894))

minus1

which plays an importantrole in our error estimates In this paper we assume 119892(r) gt

1119901 which ensures that 119861r119901120579(R119889

) is embedded into 119862(R119889) by

a Sobolev-type embedding theorem and therefore functionvalues are well defined see [20]

Now we make some illustrations about why we chooseBesov spaces as the hypothesis function spaces that is whywe assume the signals come from Besov spaces Firstly instudying the aliasing errors for nonbandlimited functionsone often uses Lipschitz or Sobolev regularity to replacethe strong bandlimited assumption In this way one canderive some reasonable convergence rates as the distancebetween the sampling periods tends to zero However thealiasing and truncation errors by local sampling for thesenonbandlimited functions have not been thoroughly studiedIn particular errors by localized sampling approximationfor these spaces of functions with measured values havenever been considered In this paper using the tools in thestudy of mean 120590-dimension width for Besov classes and therelated imbedding theorems we can consider anisotropicBesov spaces which include Lipschitz or Sobolev spacesas special cases Thus our results immediately lead to theresults for these two types of hypothesis function spaces Ofcourse the results on these two spaces are also novel Onthe other hand from the viewpoint of approximation theoryit is worth studying the Besov class since the best possibleorders of approximation by bandlimited functions are knownfor Besov classes from the corresponding results of mean 120590-dimension width theory Note that a convergent Shannonseries is a bandlimited function So it is natural to ask if onecan use Shannon interpolation formula to realize the bestapproximation for these spaces In what follows we will givean affirmative answer to this question

By the way for later use we recall the classical Sobolevspace 119882

119903

119901(R119889

) which consists of functions 119891 isin 119871119901(R119889) such

that for all multi-index vector l = (1198971 119897119889) isin N119889 with|l| = sum

119889

119895=1119897119895 le 119903 the distributional partial derivative

120597l119891 =

120597|l|119891

12059711989711199091 sdot sdot sdot 120597119897119889119909119889

(10)

belongs to 119871119901(R119889)

The remaining part of this paper is organized as followsIn Section 2 we consider errors in truncated Shannon sam-pling series with exactly functional values based on localizedsampling In Section 3 we firstly generalize part of the resultsin Section 2 to the sampling series with measured sampledvalues and then give some applications

In what follows let k t and so forth denote vectorvariables living inR119889 andwrite kk = (1198961V1 119896119889V119889) andk sdott = (V11199051 V119889119905119889) We use the same symbol119862 for possiblydifferent positive constants These constants are independentof N isin N119889 and k isin R119889

+ Denote by [119909] the largest integer not

exceeding 119909

2 The Exactly Functional Values Case

The famous Shannon sampling theorem states every function119891 isin 119861

2

V(R) can be completely reconstructed from its sampled

Abstract and Applied Analysis 3

values taken at instances 119896V119896isinZ (cf [1]) In this case therepresentation of 119891 is given by

119891 (119905) = (119878V119891) (119905) =

+infin

sum

119896=minusinfin

119891(119896

V) sinc (V119905 minus 119896) (11)

where sinc (119905) = sin120587119905120587119905 119905 = 0 and sinc (0) = 1 Series (11)converges absolutely and uniformly on R

In [10] the authors establish multidimensional Shannonsampling theorem by extending (11) to the case 119891 isin 119861

119901

k (R119889)

1 lt 119901 lt infin and 119889 gt 1 They obtained the following theorem

TheoremA Let119891 isin 119861119901

k (R119889) 1 le 119901 lt infinThen for any t isin R119889

119891 (t) = (119878k119891) (t) = sum

kisinZ119889119891(

kk) sinc (k sdot t minus k) (12)

where sinc (t) = prod119889

119894=1sinc(119905119894) The series on the right-hand

side of (12) converges absolutely and uniformly on R119889

Shannonrsquos expansion requires us to know the exact valuesof a signal 119891 at infinitely many points and to sum an infiniteseries In practice only finitely many samples are availableand hence the symmetric truncation error

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (t) minus sum

|119896119889|le119873119889

sdot sdot sdot sum

|1198961|le1198731

119891(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(13)

has been widely studied under the assumption that119891 satisfiessome decay condition Among others in [11] the uniformtruncation error bounds are determined for 119891 isin 119861

2

V(R) witha decay condition In [12] the uniform bounds of truncationerror and aliasing error are derived for functions belongingto the Besov class 119861r

infin120579(R119889

) with the same decay condition asin [11] Since their results are the motivations of our workswe restate them as follows Throughout the paper we denotethe unit ball of the space 119861r

119901120579(R119889

) byU(119861r119901120579(R119889

))

Theorem B (see [12]) Let 119891 isin U(119861rinfin120579

(R119889)) 1 le 120579 le infin and

r isin R119889

+satisfy the decay condition inequality

1003816100381610038161003816119891 (t)1003816100381610038161003816 le119860

(1 + |t|2)120575 (14)

where 119860 gt 0 and 0 lt 120575 le 1 are constants and |t|2 = (1199052

1+

sdot sdot sdot + 1199052

119889)12 For 120590 gt 0 define the associated k = (V1 V119889) by

setting V119895 = 120590119892(r)119903119895 for 119895 isin 119885119889 If V119894 gt (12)119890

2120575 for 119894 isin 119885119889 then

1003816100381610038161003816119891 (t) minus (119878k119891) (t)1003816100381610038161003816 le 119862 sdot 120590

minus119892(r)ln119889120590 (15)

Theorem C (see [12]) Let 119891 isin U(119861rinfin120579

(R119889)) 1 le 120579 le

infin satisfy the decay condition (14) Then for any N =

(1198731 119873119889) isin N119889 with119873119894 gt (12)1198902120575 119894 = 1 119889 one has

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (t) minus sum

|119896119889|le119873119889

sdot sdot sdot sum

|1198961|le1198731

119891(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862(

119889

sum

119894=1

ln119873119894)

119889119889

sum

119894=1

119873minus119903119894(1+(119899120575)119903119894)

119894

(16)

Now we truncate the series on the right-hand side of (12)based on localized sampling That is if we want to estimate119891(t) we only sum over values of 119891 on a part of Z119889

k near tThus for any N isin N119889 we consider the finite sum

(119878kN119891) (t) = sum

kminusksdottisin119868119889N

119891(kk) sinc (k sdot t minus k) (17)

as an approximation to 119891(t) In this way we can derive theuniform bounds for the associated truncation error

(119864kN119891) (t) =1003816100381610038161003816119891 (t) minus (119878kN119891) (t)

1003816100381610038161003816 (18)

and aliasing error1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 (19)

without any assumption about the decay of 119891 isin U(119861r119901120579(R119889

))Our main result of this section is the following uniform

bound of the aliasing error1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 (20)

Theorem 1 Let 119891 isin U(119861r119901120579(R119889

)) with 1 lt 119901 lt infin 1 le 120579 le

infin and 119903119894 gt 119889 for 119894 isin 119885119889 For 120590 gt 119890 define k in the samemanner as in Theorem B then one has

1003816100381610038161003816119891 (t) minus (119878k119891) (t)1003816100381610038161003816 le 119862 sdot 120590

minus119892(r)+1119901ln119889120590 (21)

We firstly note that due to the localized sampling thefunction in Theorem 1 does not need to satisfy any decayassumption at infinity Next we make a comment on thebound 120590

minus119892(r)+1119901ln119889120590 It is known from the results of mean120590-dimension Kolmogorovwidths for Besov classU(119861

r119901120579(R119889

))

that

inf119892isin119861119901

k (R119889)

sup119891isinU(119861r

119901120579(R119889))

1003817100381710038171003817119891 minus 1198921003817100381710038171003817119871infin

ge 119862120590minus119892(r)+1119901

(22)

Thus the bound inTheorem 1 is optimal up to the logarithmicfactor ln119889120590 see [21] As a consequence ofTheorem 1 we showthat using truncated sampling series (17) we can still achievethis near optimal bound

4 Abstract and Applied Analysis

Theorem 2 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 and ras in Theorem 1 For 120590 gt 119890 define k as in Theorem B Then forN = (1198731 119873119889) isin N119889 with119873119894 = [(120590

119892(r))119901

+ 1] one has

(119864kN119891) (t) le 119862 sdot 120590minus119892(r)+1119901ln119889120590 (23)

To prove Theorem 1 we will choose an intermediatefunction which is a good approximation for both 119891 and 119878k119891Now we describe how to choose this function For moredetails one can see [21 22]

For any positive real number119906 gt 0 we define the function

119892119906 (119909) = 119860 119904(sinc120587minus1119906119909)

2119904 119909 isin R 2119904 gt 1 (24)

where the constant 119860 119904 is taken such that intR119892119906(119909)119889119909 = 1

Suppose 119906119895 gt 0 119895 isin 119885119889 For any 119891 isin 119861r119901120579

(R119889) set

(119879119904

119906119895119891) (t)

= intR

119892119906119895(119909119895) ((minus1)

119897119895+1 (Δ119897119895119909119895119891) (t) + 119891 (t)) 119889119909119895

= intR

119892119906119895(119909119895)

119897119895

sum

119894=1

119889119894119891 (1199051 119905119895minus1 119905119895 + 119894119909119895 119905119895+1 119905119889) 119889119909119895

(25)

where sum119897119895

119894=1119889119894 = 1

When 119905 isin R and 119895 isin 119885119889 we let

119866119906119895(119909) =

119897119895

sum

119894=1

119889119894

119894119892119906119895

(119909

119894) (26)

and observe from formulas (25) and (26) that 119879119906119895has the

alternative representation

(119879119904

119906119895119891) (t) = int

R

119866119906119895(119909119895) 119891 (t + 119894119909119895e119895) 119889119909119895 x isin R

119889 (27)

We define the value of a kernel 119866u at x = (1199091 1199092 119909119889) by

119866u (x) = prod

119894isin119885119889

119866119906119894(119909119894) (28)

and introduce the operator

119879119904

u = 119879119904

1199061∘ sdot sdot sdot ∘ 119879

119904

119906119889 (29)

Consequently 119879119904

u is given by

(119879119904

u119891) (t) = intR119889

119866u (x) 119891 (t + x) 119889x t isin R119889 (30)

It is known from [20] that 119879119904

u119891 isin 119861119901

2119904u(R119889) We will exploit

the following properties of 119879119904

u119891 in the proof of Theorem 1

Lemma3 Let119891 isin U(119861r119901120579(R119889

)) 1 le 119901 and 120579 le infin For120590 gt 0define u isin R119889 with 119906119895 = 120590

119892(r)119903119895 for 119895 isin 119885119889 then one has

1003817100381710038171003817119891 minus 119879119904

u1198911003817100381710038171003817119871infin

le 119862 sdot 120590minus119892(r)+1119901

(31)

Proof When 119901 = infin the inequality was proved in [12] Bythe imbedding relationship

U (119861r119901120579

(R119889)) sub 119862 sdotU (119861

r1015840infin120579

(R119889)) (32)

where r1015840 = (1minus(1119901)(sum119889

119895=1(1119903119895)))r (see [20] formore details)

we can derive the corresponding inequalities for the case 1 le

119901 lt infin from that of 119901 = infin

Lemma 4 (see [22]) If 119891 isin 119871119901(R119889) 1 le 119901 le infin and u isin R119889

+

then

1003817100381710038171003817119879119904

u1198911003817100381710038171003817119871119901

le 21205721003817100381710038171003817119891

1003817100381710038171003817119871119901 (33)

where 120572 = sum119894isin119885119889119897119894

For 1 le 119901 lt infin let 119897119901(Z119889) be the Banach space of all infinite

bounded 119901-summable sequences y = 119910kkisinZ119889 such that thenorm

1003817100381710038171003817y1003817100381710038171003817119897119901

= ( sum

kisinZ119889|119910k|

119901)

1119901

(34)

is finite

Lemma 5 (see [10]) Let y = 119910k isin 119897119901(Z119889

) 1 lt 119901 lt infin Thenthe series

119871k (y t) = sum

kisinZ119889119910k sinc (k sdot t minus k) (35)

converges uniformly on R119889 to a function in 119861119901

k (R119889)

We also need the following bound for sinc seriessumkisinZ119889 |sinc(k sdot t minus k)|119902

Lemma 6 (see [11]) Let 119889 ge 1 119902 gt 1 k = (V1 V119889) andV119894 gt 1 119894 = 1 119889 Then for any t isin R119889

( sum

kisinZ119889|sinc (k sdot t minus k)|119902)

1119902

le (119902

119902 minus 1)

119889

(36)

For 119891 isin 119861119901

k (R119889) one has the following Marcinkiewicz-type

inequality

Lemma 7 (see [20 23]) Let 119891 isin 119861119901

k (R119889) 1 le 119901 lt infin Then

one has

(

119889

prod

119894=1

Vminus1119894

sum

kisinZ119889

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901 (37)

The next lemma presents a Marcinkiewicz-type inequal-ity for functions from Sobolev spaces

Abstract and Applied Analysis 5

Lemma 8 (see [10]) Let 119891 isin 119882119897

119901(R119889

) 1 le 119901 lt infin and 119897 ge 119889Then

(

119889

prod

119894=1

1

V119894sum

kisinZ119889

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

le 119862(1003817100381710038171003817119891

1003817100381710038171003817119871119901+

119889

sum

119894=1

1

V119894

10038171003817100381710038171003817100381710038171003817

120597119891

120597119909119894

10038171003817100381710038171003817100381710038171003817119871119901

+ sum

1le119894le119895le119889

1

V119894

1

V119895

1003817100381710038171003817100381710038171003817100381710038171003817

1205972119891

120597119909119894120597119909119895

1003817100381710038171003817100381710038171003817100381710038171003817119871119901

+ sdot sdot sdot +

119889

prod

119894=1

1

V119894

100381710038171003817100381710038171003817100381710038171003817

120597119889119891

1205971199091 sdot sdot sdot 120597119909119889

100381710038171003817100381710038171003817100381710038171003817119871119901

)

(38)

Lemma 9 (see [20 24]) Let l = (1198971 119897119889) isin N119889 r =

(1199031 119903119889) isin R119889

+ 1 le 119901 120579 le infin 1 minus sum

119889

119894=1119897119894119903119894 gt 0 and

r1015840 = (1 minus sum119889

119894=1119897119894119903119894)r For 119891 isin 119861

r119901120579(R119889

) it follows that thereexists a constant119862 depending on r r1015840 119901 and 120579 but independentof 119891 such that

10038171003817100381710038171003817120597|l|11989110038171003817100381710038171003817119861r1015840119901120579

le 1198621003817100381710038171003817119891

1003817100381710038171003817119861r119901120579

(39)

Proof of Theorem 1 It is known from Lemma 9 (letting 119897119894 = 1

for 119894 isin 119885119889) that the fact 119891 isin U(119861rp120579(R

119889)) with 119903119894 gt 119889 119894 isin 119885119889

implies 119891 isin 119882119889

119901(R119889

) therefore by Lemma 810038171003817100381710038171003817100381710038171003817

119891(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

1119901 (40)

And hence by Lemma 5

(119878k119891) (t) = sum

kisinZ119889119891(

kk) sinc (k sdot t minus k) (41)

converges uniformly on R119889Set u = k2119904 and 2119904 gt 119889 +max119903119894 119894 = 1 119889 So 119879

119904

u119891 isin

119861119901

k (R119889) as mentioned above By Theorem A we have 119879

119904

u119891 =

119878k(119879119904

u119891) Thus

(119879119904

u119891) (t) minus (119878k119891) (t)

= sum

kisinZ119889((119879

119904

u119891)(kk) minus 119891(

kk)) sinc (k sdot t minus k)

(42)

Using the triangle inequality we obtain1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 +

1003816100381610038161003816(119879119904

u119891) (t) minus (119878k119891) (t)1003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 +

1003816100381610038161003816119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

+1003816100381610038161003816119878k (119879

119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

(43)

By Lemma 31003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 le 119862120590

minus119892(r)+1119901 (44)

It is clear that1003816100381610038161003816119878kN (119879

119904

u119891 minus 119891) (t)1003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminuskisin119868119889N

(119879119904

u119891 minus 119891)(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(45)

Applying Holderrsquos inequality with exponent 1199010 we get1003816100381610038161003816119878kN (119879

119904

u119891 minus 119891) (t)1003816100381610038161003816

le ( sum

ksdottminuskisin119868119889N

10038161003816100381610038161003816100381610038161003816

(119879119904

u119891 minus 119891)(kk)

10038161003816100381610038161003816100381610038161003816

1199010

)

11199010

sdot ( sum

ksdottminuskisin119868119889N

|sinc(k sdot t minus k)|1199020)11199020

le 119862(

119889

prod

119894=1

2119873119894 + 1)

11199010

120590minus119892(r)+1119901

sdot 119901119889

0

(46)

where 11199010+11199020 = 1 and the second inequality follows from(44) and Lemma 6

Next we estimate |119878k(119879119904

u119891 minus 119891)(t) minus 119878kN(119879119904

u119891 minus 119891)(t)| ByHolderrsquos inequality

1003816100381610038161003816119878k (119879119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

le ( sum

ksdottminusknotin119868119889N

10038161003816100381610038161003816100381610038161003816

(119879119904

u119891 minus 119891)(kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

sdot ( sum

ksdottminusknotin119868119889N

100381610038161003816100381610038161003816sinc(k sdot t minus k)

100381610038161003816100381610038161003816

119902

)

1119902

(47)

where 1119901 + 1119902 = 1By Lemma 7 and Lemma 4 we obtain

10038171003817100381710038171003817100381710038171003817

(119879119904

u119891)(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

11199011003817100381710038171003817119879119904

u1198911003817100381710038171003817119871119901

le 11986212059011199011003817100381710038171003817119891

1003817100381710038171003817119871119901le 119862120590

1119901

(48)

It follows from (40) (48) and Minkowski inequality10038171003817100381710038171003817100381710038171003817

(119879119904

u119891 minus 119891)(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

1119901 (49)

Set ℎ(t) = (sumksdottminusknotin119868119889N|sinc(k sdot t minus k)|119902)1119902 Note that ℎ(t +

mk) = ℎ(t) for all t isin R119889 and m isin Z119889 Thus to give anupper estimate for ℎ(t) on R119889 we only need to bound it onprod

119889

119894=1[0 1V119894] Note that

k k notin 119868119889

N sub

119889

⋃

119894=1

k 119896119894 notin [minus119873119894 119873119894] (50)

6 Abstract and Applied Analysis

A straightforward computation shows that for 119905119894 isin [0 1V119894]

( sum

119896119894notin(minus119873119894 119873119894]

|sinc (V119894119905119894 minus 119896119894)|119902)

1119902

le (119862 sum

119896119894notin(minus119873119894 119873119894]

1

|119896|119902)

1119902

le (119862int

infin

119873119894

119905minus119902)

1119902

le 119862119873minus1119901

119894

(51)

Therefore

1003816100381610038161003816119878k (119879119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816 le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894)

(52)

It follows from (46) and (52) that

1003816100381610038161003816(119879119904

u119891 minus 119878k119891) (t)1003816100381610038161003816

le 119862((

119889

prod

119894=1

(2119873119894 + 1))

11199010

120590minus119892(r)+1119901

119901119889

0

+1205901119901

(

119889

sum

119894=1

119873minus1119901

119894))

(53)

We choose119873119894 = [(120590119892(r)

)119901

+ 1] and 1199010 = sum119889

119894=1ln(2119873119894 + 1) It is

easy to see that 1199010 gt 1 and (prod119889

119894=1(2119873119894 + 1))

11199010= 119890 A simple

computation gives

119901119889

0le 119862(ln

119889

prod

119894=1

119873119894)

119889

le 119862 ln119889120590 (54)

Note that119873119894 ge 120590119892(r)119901 Thus we have

119889

sum

119894=1

119873minus1119901

119894le 119862120590

minus119892(r) (55)

Collecting the above results we obtain

1003816100381610038161003816(119879119904

u119891) (t) minus (119878k119891) (t)1003816100381610038161003816 le 119862 sdot 120590

minus119892(r)+1119901ln119889120590 (56)

Combining (44) and (56) we prove the theorem

Proof of Theorem 2 By the triangle inequality we have

(119864kN119891) (t) le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816

+1003816100381610038161003816(119878k119891) (t) minus (119878kN119891) (t)

1003816100381610038161003816

(57)

By the arguments similar to those used in the proof ofTheorem 1 we obtain

1003816100381610038161003816(119878k119891) (t) minus (119878kN119891) (t)1003816100381610038161003816

le ( sum

ksdottminusknotin119868119889N

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

sdot ( sum

ksdottminusknotin119868119889N

100381610038161003816100381610038161003816sinc (k sdot t minus k)

100381610038161003816100381610038161003816

119902

)

1119902

le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894)

le 119862120590minus119892(r)+1119901ln119889120590

(58)

where we use119873119894 = [(120590119892(r)

)119901

+ 1] in the last inequalityCombiningTheorem 1 and (58) we complete the proof of

Theorem 2

3 The Measured Sampled Values Case

In practice the sampled values of a signal may not be exactlythe functional values and may have to be quantized Typicalerrors arising from these facts are jitter errors and amplitudeerrors Using the key idea of quasi-interpolation whichadopts integer translations of a basic function and integertranslations of a linear functional to approximate functionssee [8 25] and the references therein We may considersampled values that are the results of a linear functional andits integer translations acting on an undergoing signal [4 25]Such sampled values are called measured sampled valuesbecause they are closer to the truemeasurements taken fromasignalThe sampling series with themeasured sampled valuesis defined to be

(119878120582

k119891) (t) = sum

kisinZ119889120582119896119891(sdot +

kk) sinc (k sdot t minus k) (59)

where 120582 = 120582119896119896isinZ is any sequence of continuous linearfunctionals 1198620(R

119889) rarr C with 1198620(R

119889) being the set of all

continuous functions defined on R119889 and tending to zero atinfinity

Similar to the definition of (119878kN119891)(t) and (119864kN119891)(t) wehave the finite sum

(119878120582

kN119891) (t) = sum

kminusksdottisin119868119889N

120582119896119891(sdot +kk) sinc (k sdot t minus k) (60)

and the truncation error

(119864120582

kN119891) (t) =10038161003816100381610038161003816119891 (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816 (61)

To establish our theorems we need the error modulus

Ωk (119891 120582) = sup119896isinZ

10038161003816100381610038161003816100381610038161003816

120582119896119891(sdot +kk) minus 119891(

kk)

10038161003816100381610038161003816100381610038161003816

k gt 0 (62)

Abstract and Applied Analysis 7

WewriteΩ(119891 120582) forΩk(119891 120582) if no confusion arisesThe errormodulusΩ(119891 120582) provides a quantity for the quality of signalrsquosmeasured sampling values When the functionals in 120582 areconcrete we may get some reasonable estimates forΩ(119891 120582)

Sampling series with measured sampled values has beenstudied in [8] for bandlimited functions but without trunca-tionThe truncation errors are considered for functions fromLipschitz class with a decay condition in [13] Now we recalla typical result in [13]

Denote by Lip119871(1 119862(R119889

)) the set of all continuous func-tions 119891 satisfying

1003816100381610038161003816119891 (x) minus 119891 (y)1003816100381610038161003816 le 119871 (10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119909119889 minus 119910119889

1003816100381610038161003816) (63)

for all x = (1199091 119909119889) and y = (1199101 119910119889) isin R119889 SetkV = (1198961V 119896119889V) V sdot t = (V1199051 V119905119889) and tinfin =

max1199051 119905119889

Theorem D Let 119891 isin Lip119871(1 119862(R119889

)) satisfy the decay condi-tion

1003816100381610038161003816119891 (t)1003816100381610038161003816 le 119872119891tminus120574

infin tinfin ge 1 (64)

for some 0 lt 120574 le 1 Let 120582 = 120582119896 be any sequence of continuouslinear functionals For each V ge 119890

2 one has for the truncationerror at t isin R119889

10038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (t) minus sum

kisinZ119889kle119873120582119896119891(

kV) sinc (V sdot t minus k)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862Vminus1ln119889V

(65)

provided119873 = lfloor(V1+1120574minus1)2rfloor where lfloor119909rfloor is the smallest integerthat is greater or equal to a given 119909 isin R and Ω(119891 120582) le 1198880V

minus1

In [9] the author obtain the uniform bound of symmetrictruncation error for functions from isotropic Besov spacewith a similar decay condition Now we will provide theestimation for the truncation error

(119864120582

kN119891) (t) =10038161003816100381610038161003816119891 (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816

(66)

without any assumption about the decay of 119891 isin 119861r119901120579(R119889

)

Theorem 10 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define k as in Theorem B Let120582 = 120582119896 be any sequence of continuous linear functionals Fork one has

(119864120582

kN119891) (t) le 119862 sdot 120590minus119892(r)+1119901ln119889120590 (67)

provided 119873119894 = [(120590119892(r)

)119901

+ 1] and Ω(119891 120582) le 1198620120590minus119892(r)+1119901 for

some constant 1198620 gt 0

Proof By the triangle inequality we have

(119864120582

kN119891) (t)

le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 +10038161003816100381610038161003816(119878k119891) (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 + 1198681 + 1198682

(68)

where

1198681 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminusknotin119868119889N

119891(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198682 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminuskisin119868119889N

(119891(kk) minus 120582119896119891(sdot +

kk)) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)

Similar to (52) we have

1198681 le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894) (70)

Using Holderrsquos inequality we obtain

1198682 le ( sum

ksdottminuskisin119868119889N

10038161003816100381610038161003816100381610038161003816

119891 (kk) minus 120582119896119891(sdot +

kk)

10038161003816100381610038161003816100381610038161003816

1199010

)

11199010

sdot ( sum

ksdottminuskisin119868119889N

|sinc (k sdot t minus k) |1199020)11199020

le 119862(

119889

prod

119894=1

(2119873119894 + 1))

11199010

Ω(119891 120582) sdot 119901119889

0

(71)

where 11199010+11199020 = 1 Nowwe select the same119873119894 and1199010 as inthe proof ofTheorem 1 Similar to (55) we havesum119889

119894=1119873

minus1119901

119894le

119862120590minus119892(r) Thus

1198681 le 119862120590minus119892(r)+1119901

(72)

A simple computation gives (prod119889

119894=1(2119873119894 + 1))

11199010= 119890 and 1199010 le

119862 lnprod119889

119894=1119873119894 le 119862 ln120590 Notice that Ω(119891 120582) le 1198620120590

minus119892(r)+1119901Collecting these results we obtain

1198682 le 119862120590minus119892(r)+1119901ln119889120590 (73)

It follows fromTheorem 1 (72) and (73) that

(119864120582

kN119891) (t) le 119862120590minus119892(r)+1119901ln119889120590 (74)

which completes the proof

Finally we apply Theorem 10 to some practical examplesThe first one is that the measured sampled values are given byaverages of a function For119891 isin 1198620(R

119889)wedefine themodulus

of continuity

120596 (119891 120591) = suphle120591

1003817100381710038171003817119891 (sdot + h) minus 119891 (sdot)1003817100381710038171003817infin

(75)

where 120591may be any positive number

8 Abstract and Applied Analysis

Corollary 11 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define kN as in Theorem 10Suppose the sampled values 119891k of 119891 are obtained by the rule

119891k =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119891(t + kk)119889t (76)

where ℎ119895 are numbers satisfying 0 lt ℎ119895 le 120591 for all k isin Z119889

and 120591 gt 0 If 120591 lt 12 and 120596(119891 120591) le min119890119889(minus119892(r)+1119901)1198620120590

minus119892(r)+1119901 then

(119864120582

kN119891) (t) le 119862 sdot 120596(119891 120591)119892(r)minus1119901ln119889 1

120596 (119891 120591) (77)

Proof Let 120582 = 120582119896119896isinZ be the sequence of the linearfunctionals on 1198620(R

119889) and 119892 a continuous function on 119868

119889

h Define

120582119896119892 =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119892(t + kk)119889t (78)

Then 119891k = 120582119896119891(sdot + kk) Clearly |119891(t + kv) minus 119891(kv)| le

120596(119891 120591) Therefore

Ωk (119891 120582)

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816

120582119896119891(t + kk) minus 119891(

kk)

10038161003816100381610038161003816100381610038161003816

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

(119891(t + kk) minus 119891(

kk))119889t

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

120596 (119891 120591) 119889t10038161003816100381610038161003816100381610038161003816100381610038161003816

= 120596 (119891 120591)

(79)

Note that the function 119909 997891rarr 119909119892(r)minus1119901ln119889(1119909) is monotonely

increasing for 119909 isin (0 119890119889(minus119892(r)+1119901)

) The corollary followsfromTheorem 10

The second example is an estimate for the combination ofall four errors existing in sampling series the amplitude errorthe time-jitter error the truncation errors and the aliasingerrors We give some explanation for the amplitude error andthe time-jitter error

We assume the amplitude error results from quantizationwhich means the functional value 119891(119905) of a function 119891 atmoment 119905 is replaced by the nearest discrete value ormachinenumber 119891(119905) The quantization size is often known beforehand or can be chosen arbitrarily We may assume that thelocal error at any moment 119905 is bounded by a constant 120576 gt 0that is |119891(119905) minus 119891(119905)| le 120576 The time-jitter error arises if thesampled instances are notmet correctly butmight differ fromthe exact ones by 120591

1015840

119896 119896 isin Z we assume |1205911015840

119896| le 120591

1015840 for all 119896 andsome constant 1205911015840 gt 0 The combined error is defined to be

(119864119873119891) (119905) = 119891 (119905) minus sum

Vsdot119905minus119896isin[minus119873119873]

119891(119896

V+ 120591

1015840

119896) sinc (V119905 minus 119896)

(80)

Corollary 12 Let 119891 isin U(119861119903

119901120579(R)) 1 lt 119901 lt infin 1 le 120579 le infin

119903 gt 1 and V gt 119890 Then1003817100381710038171003817119864119873119891

1003817100381710038171003817infinle 119862Vminus119903+1119901 ln V (81)

provided119873 = [V119903119901+1] |119891(119905)minus119891(119905)| le 1198881Vminus119903+1119901 and120596(119891 120591

1015840) le

1198882Vminus119903+1119901 where 1198881 1198882 are positive constants and 1198881 + 1198882 le 1198620

Proof We define

120582119896 =

119891 (119896V + 1205911015840

119896)

119891 (119896V + 1205911015840

119896)120575 (sdot minus 120591

1015840

119896) (82)

where 120575 is the Dirac distribution Then 120582 = 120582119896119896isinZ is asequence of linear functional on1198620(R) It is clear that 120582119896119891(sdot+119896V) = 119891(119896V + 120591

1015840

119896) Then

10038161003816100381610038161003816100381610038161003816

120582119896119891(sdot +119896

V) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V+ 120591

1015840

119896)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le (1198881 + 1198882) Vminus119903+1119901

(83)

Thus Ω(119891 120582) le 1198620Vminus119903+1119901 By Theorem 10 we get the desired

result

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 10971251 11101220 and11271199) and the Program forNewCentury Excellent Talentsat University of China (NCET-10-0513)

References

[1] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 1948

[2] P L ButzerW Engels and U Scheben ldquoMagnitude of the trun-cation error in sam-pling expansions of bandlimited signalsrdquoIEEE Transactions on Acoustics Speech and Signal Processingvol 30 no 6 pp 906ndash912 1982

[3] A I ZayedAdvances in Shannonrsquos SamplingTheory CRC PressBoca Raton Fla USA 1993

[4] SDCasey andD FWalnut ldquoSystems of convolution equationsdeconvolution Shannon sampling and the wavelet and Gabortransformsrdquo SIAM Review vol 36 no 4 pp 537ndash577 1994

[5] P L Butzer G Schmeisser and R L Stens ldquoAn introductionto sampling analysisrdquo in Nonuniform Sampling Theory andPractice F Marvasti Ed pp 17ndash121 Kluwer Academic NewYork NY USA 2001

[6] S Smale and D-X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquo Bulletin of the AmericanMathematical Society vol 41 no 3 pp 279ndash305 2004

[7] S Smale and D-X Zhou ldquoShannon sampling II Connectionsto learning theoryrdquo Applied and Computational HarmonicAnalysis vol 19 no 3 pp 285ndash302 2005

Abstract and Applied Analysis 9

[8] P L Butzer and J Lei ldquoApproximation of signals usingmeasuredsampled values and error analysisrdquo Communications in AppliedAnalysis vol 4 no 2 pp 245ndash255 2000

[9] P X Ye ldquoError analysis for Shannon sampling series approxima-tionwithmeasured sampled valuesrdquoResearch Journal of AppliedSciences Engineering and Technology vol 5 no 3 pp 858ndash8642013

[10] J J Wang and G S Fang ldquoA multidimensional sampling theo-rem and an estimate of the aliasing errorrdquo Acta MathematicaeApplicatae Sinica vol 19 no 4 pp 481ndash488 1996

[11] X M Li ldquoUniform bounds for sampling expansionsrdquo Journal ofApproximation Theory vol 93 no 1 pp 100ndash113 1998

[12] L Jingfan and F Gensun ldquoOn uniform truncation error boundsand aliasing error for multidimensional sampling expansionrdquoSampling Theory in Signal and Image Processing vol 2 no 2pp 103ndash115 2003

[13] P L Butzer and J Lei ldquoErrors in truncated sampling serieswith measured sampled values for not-necessarily bandlimitedfunctionsrdquo Functiones et Approximatio vol 26 pp 25ndash39 1998

[14] H D Helms and J B Thomas ldquoTruncation error of sampling-theorem expansionsrdquo Proceedings of The IRE vol 50 no 2 pp179ndash184 1962

[15] D Jagerman ldquoBounds for truncation error of the samplingexpansionrdquo SIAM Journal on Applied Mathematics vol 14 no4 pp 714ndash723 1966

[16] C A Micchelli Y Xu and H Zhang ldquoOptimal learning ofbandlimited functions from localized samplingrdquo Journal ofComplexity vol 25 no 2 pp 85ndash114 2009

[17] A Ya Olenko and T K Pogany ldquoUniversal truncation errorupper bounds in sampling restorationrdquo Georgian MathematicalJournal vol 17 no 4 pp 765ndash786 2010

[18] P-X Ye and Z-H Song ldquoTruncation and aliasing errors forWhittaker-Kotelnikov-Shannon sampling expansionrdquo AppliedMathematics B vol 27 no 4 pp 412ndash418 2012

[19] P X Ye B H Sheng and X H Yuan ldquoOptimal order oftruncation and aliasing errors formulti-dimensional whittaker-shannon sampling expansionrdquo International Journal of Wirelessand Mobile Computing vol 5 no 4 pp 327ndash333 2012

[20] S M Nikolskii Approximation of Functions of Several Variablesand Imbedding Theorems Springer New York NY USA 1975

[21] Y Jiang and Y Liu ldquoAverage widths and optimal recovery ofmultivariate Besov classes in 119871119901(119877

119889)rdquo Journal of Approximation

Theory vol 102 no 1 pp 155ndash170 2000[22] C A Micchelli Y S Xu and P X Ye ldquoCucker-Smale learning

theory in Besov spacesrdquo in Advances in LearnIng TheoryMethods Models and Applications J Suykens G HorvathS Basu et al Eds pp 47ndash68 IOS Press Amsterdam TheNetherlands 2003

[23] R P Boas Jr Entire Functions Academic Press New York NYUSA 1954

[24] G Fang F J Hickernell and H Li ldquoApproximation onanisotropic Besov classes with mixed norms by standard infor-mationrdquo Journal of Complexity vol 21 no 3 pp 294ndash313 2005

[25] H G Burchard and J Lei ldquoCoordinate order of approxima-tion by functional-based approximation operatorsrdquo Journal ofApproximation Theory vol 82 no 2 pp 240ndash256 1995

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

exponential type k provided that for every 120576 gt 0 there exists apositive number 119888 such that for all complex vectors z = (119911119895

119895 isin 119885119889) isin C119889 we have the bound

|ℎ (z)| le 119888 exp( sum

119895isin119885119889

(V119895 + 120576)10038161003816100381610038161003816119911119895

10038161003816100381610038161003816) (3)

Denote by 119864k(C119889) the space of all entire functions of expo-

nential type k Let 119861k(R119889) be the subset of 119864k(C

119889) which are

bounded on R119889 Set

119861119901

k (R119889) = 119861k (R

119889) cap 119871119901 (R

119889) 1 le 119901 le infin (4)

Every vector k = (V119895 119895 isin 119885119889) isin R119889

+determines the rectangle

119868119889

k = prod

119895isin119885119889

[minusV119895 V119895] (5)

According to the Schwartz theorem [20]

119861119901

k (R119889) = 119891 isin 119871119901 (R

119889) supp 119891 sube 119868

119889

k (6)

where 119891 is the Fourier transform of 119891 in the sense ofdistribution For the case 119901 = 2 it is the classical Paley-Wiener theorem

Now we define anisotropic Besov space Suppose that 119896 isin

119873 and t isin R119889 For 119891 isin 119871119901(R119889) we define the 119896th partial

difference of 119891 in the 119897th coordinate direction e119897 at the pointt isin R119889 with step 119909119897 isin R by the formula

Δ119896

119909119897119891 (t) =

119896

sum

119894=0

(minus1)119894+119896

(119896

119894)119891 (t + 119894119909119897e119897) (7)

Let l = (1198971 119897119889) isin N119889 r = (1199031 119903119889) isin R119889

+ and 119897119894 gt 119903119894 for

119894 isin 119885119889 1 le 119901 and 120579 le infinWe say119891 isin 119861r119901120579(R119889

) if119891 isin 119871119901(R119889)

and the following seminorm is finite

10038171003817100381710038171198911003817100381710038171003817119887119903119894

119901120579

=

intR

(

10038171003817100381710038171003817Δ11989711989411990911989411989110038171003817100381710038171003817119871119901

|119909119894|119903119894+(1120579)

)

120579

119889119909119894

1120579

1 le 120579 lt infin

sup|119909119894| = 0

10038171003817100381710038171003817Δ11989711989411990911989411989110038171003817100381710038171003817119871119901

|119909119894|119903119894

120579 = infin

119894 = 1 119889

(8)

The linear space 119861r119901120579(R119889

) is a Banach space with the norm

10038171003817100381710038171198911003817100381710038171003817119861r119901120579

=1003817100381710038171003817119891

1003817100381710038171003817119871119901+ sum

119894isin119885119889

10038171003817100381710038171198911003817100381710038171003817119887119903119894

119901120579 (9)

and is called an anisotropic Besov space We introduce thenotation 119892(r) = (sum

119889

119894=1(1119903119894))

minus1

which plays an importantrole in our error estimates In this paper we assume 119892(r) gt

1119901 which ensures that 119861r119901120579(R119889

) is embedded into 119862(R119889) by

a Sobolev-type embedding theorem and therefore functionvalues are well defined see [20]

Now we make some illustrations about why we chooseBesov spaces as the hypothesis function spaces that is whywe assume the signals come from Besov spaces Firstly instudying the aliasing errors for nonbandlimited functionsone often uses Lipschitz or Sobolev regularity to replacethe strong bandlimited assumption In this way one canderive some reasonable convergence rates as the distancebetween the sampling periods tends to zero However thealiasing and truncation errors by local sampling for thesenonbandlimited functions have not been thoroughly studiedIn particular errors by localized sampling approximationfor these spaces of functions with measured values havenever been considered In this paper using the tools in thestudy of mean 120590-dimension width for Besov classes and therelated imbedding theorems we can consider anisotropicBesov spaces which include Lipschitz or Sobolev spacesas special cases Thus our results immediately lead to theresults for these two types of hypothesis function spaces Ofcourse the results on these two spaces are also novel Onthe other hand from the viewpoint of approximation theoryit is worth studying the Besov class since the best possibleorders of approximation by bandlimited functions are knownfor Besov classes from the corresponding results of mean 120590-dimension width theory Note that a convergent Shannonseries is a bandlimited function So it is natural to ask if onecan use Shannon interpolation formula to realize the bestapproximation for these spaces In what follows we will givean affirmative answer to this question

By the way for later use we recall the classical Sobolevspace 119882

119903

119901(R119889

) which consists of functions 119891 isin 119871119901(R119889) such

that for all multi-index vector l = (1198971 119897119889) isin N119889 with|l| = sum

119889

119895=1119897119895 le 119903 the distributional partial derivative

120597l119891 =

120597|l|119891

12059711989711199091 sdot sdot sdot 120597119897119889119909119889

(10)

belongs to 119871119901(R119889)

The remaining part of this paper is organized as followsIn Section 2 we consider errors in truncated Shannon sam-pling series with exactly functional values based on localizedsampling In Section 3 we firstly generalize part of the resultsin Section 2 to the sampling series with measured sampledvalues and then give some applications

In what follows let k t and so forth denote vectorvariables living inR119889 andwrite kk = (1198961V1 119896119889V119889) andk sdott = (V11199051 V119889119905119889) We use the same symbol119862 for possiblydifferent positive constants These constants are independentof N isin N119889 and k isin R119889

+ Denote by [119909] the largest integer not

exceeding 119909

2 The Exactly Functional Values Case

The famous Shannon sampling theorem states every function119891 isin 119861

2

V(R) can be completely reconstructed from its sampled

Abstract and Applied Analysis 3

values taken at instances 119896V119896isinZ (cf [1]) In this case therepresentation of 119891 is given by

119891 (119905) = (119878V119891) (119905) =

+infin

sum

119896=minusinfin

119891(119896

V) sinc (V119905 minus 119896) (11)

where sinc (119905) = sin120587119905120587119905 119905 = 0 and sinc (0) = 1 Series (11)converges absolutely and uniformly on R

In [10] the authors establish multidimensional Shannonsampling theorem by extending (11) to the case 119891 isin 119861

119901

k (R119889)

1 lt 119901 lt infin and 119889 gt 1 They obtained the following theorem

TheoremA Let119891 isin 119861119901

k (R119889) 1 le 119901 lt infinThen for any t isin R119889

119891 (t) = (119878k119891) (t) = sum

kisinZ119889119891(

kk) sinc (k sdot t minus k) (12)

where sinc (t) = prod119889

119894=1sinc(119905119894) The series on the right-hand

side of (12) converges absolutely and uniformly on R119889

Shannonrsquos expansion requires us to know the exact valuesof a signal 119891 at infinitely many points and to sum an infiniteseries In practice only finitely many samples are availableand hence the symmetric truncation error

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (t) minus sum

|119896119889|le119873119889

sdot sdot sdot sum

|1198961|le1198731

119891(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(13)

has been widely studied under the assumption that119891 satisfiessome decay condition Among others in [11] the uniformtruncation error bounds are determined for 119891 isin 119861

2

V(R) witha decay condition In [12] the uniform bounds of truncationerror and aliasing error are derived for functions belongingto the Besov class 119861r

infin120579(R119889

) with the same decay condition asin [11] Since their results are the motivations of our workswe restate them as follows Throughout the paper we denotethe unit ball of the space 119861r

119901120579(R119889

) byU(119861r119901120579(R119889

))

Theorem B (see [12]) Let 119891 isin U(119861rinfin120579

(R119889)) 1 le 120579 le infin and

r isin R119889

+satisfy the decay condition inequality

1003816100381610038161003816119891 (t)1003816100381610038161003816 le119860

(1 + |t|2)120575 (14)

where 119860 gt 0 and 0 lt 120575 le 1 are constants and |t|2 = (1199052

1+

sdot sdot sdot + 1199052

119889)12 For 120590 gt 0 define the associated k = (V1 V119889) by

setting V119895 = 120590119892(r)119903119895 for 119895 isin 119885119889 If V119894 gt (12)119890

2120575 for 119894 isin 119885119889 then

1003816100381610038161003816119891 (t) minus (119878k119891) (t)1003816100381610038161003816 le 119862 sdot 120590

minus119892(r)ln119889120590 (15)

Theorem C (see [12]) Let 119891 isin U(119861rinfin120579

(R119889)) 1 le 120579 le

infin satisfy the decay condition (14) Then for any N =

(1198731 119873119889) isin N119889 with119873119894 gt (12)1198902120575 119894 = 1 119889 one has

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (t) minus sum

|119896119889|le119873119889

sdot sdot sdot sum

|1198961|le1198731

119891(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862(

119889

sum

119894=1

ln119873119894)

119889119889

sum

119894=1

119873minus119903119894(1+(119899120575)119903119894)

119894

(16)

Now we truncate the series on the right-hand side of (12)based on localized sampling That is if we want to estimate119891(t) we only sum over values of 119891 on a part of Z119889

k near tThus for any N isin N119889 we consider the finite sum

(119878kN119891) (t) = sum

kminusksdottisin119868119889N

119891(kk) sinc (k sdot t minus k) (17)

as an approximation to 119891(t) In this way we can derive theuniform bounds for the associated truncation error

(119864kN119891) (t) =1003816100381610038161003816119891 (t) minus (119878kN119891) (t)

1003816100381610038161003816 (18)

and aliasing error1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 (19)

without any assumption about the decay of 119891 isin U(119861r119901120579(R119889

))Our main result of this section is the following uniform

bound of the aliasing error1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 (20)

Theorem 1 Let 119891 isin U(119861r119901120579(R119889

)) with 1 lt 119901 lt infin 1 le 120579 le

infin and 119903119894 gt 119889 for 119894 isin 119885119889 For 120590 gt 119890 define k in the samemanner as in Theorem B then one has

1003816100381610038161003816119891 (t) minus (119878k119891) (t)1003816100381610038161003816 le 119862 sdot 120590

minus119892(r)+1119901ln119889120590 (21)

We firstly note that due to the localized sampling thefunction in Theorem 1 does not need to satisfy any decayassumption at infinity Next we make a comment on thebound 120590

minus119892(r)+1119901ln119889120590 It is known from the results of mean120590-dimension Kolmogorovwidths for Besov classU(119861

r119901120579(R119889

))

that

inf119892isin119861119901

k (R119889)

sup119891isinU(119861r

119901120579(R119889))

1003817100381710038171003817119891 minus 1198921003817100381710038171003817119871infin

ge 119862120590minus119892(r)+1119901

(22)

Thus the bound inTheorem 1 is optimal up to the logarithmicfactor ln119889120590 see [21] As a consequence ofTheorem 1 we showthat using truncated sampling series (17) we can still achievethis near optimal bound

4 Abstract and Applied Analysis

Theorem 2 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 and ras in Theorem 1 For 120590 gt 119890 define k as in Theorem B Then forN = (1198731 119873119889) isin N119889 with119873119894 = [(120590

119892(r))119901

+ 1] one has

(119864kN119891) (t) le 119862 sdot 120590minus119892(r)+1119901ln119889120590 (23)

To prove Theorem 1 we will choose an intermediatefunction which is a good approximation for both 119891 and 119878k119891Now we describe how to choose this function For moredetails one can see [21 22]

For any positive real number119906 gt 0 we define the function

119892119906 (119909) = 119860 119904(sinc120587minus1119906119909)

2119904 119909 isin R 2119904 gt 1 (24)

where the constant 119860 119904 is taken such that intR119892119906(119909)119889119909 = 1

Suppose 119906119895 gt 0 119895 isin 119885119889 For any 119891 isin 119861r119901120579

(R119889) set

(119879119904

119906119895119891) (t)

= intR

119892119906119895(119909119895) ((minus1)

119897119895+1 (Δ119897119895119909119895119891) (t) + 119891 (t)) 119889119909119895

= intR

119892119906119895(119909119895)

119897119895

sum

119894=1

119889119894119891 (1199051 119905119895minus1 119905119895 + 119894119909119895 119905119895+1 119905119889) 119889119909119895

(25)

where sum119897119895

119894=1119889119894 = 1

When 119905 isin R and 119895 isin 119885119889 we let

119866119906119895(119909) =

119897119895

sum

119894=1

119889119894

119894119892119906119895

(119909

119894) (26)

and observe from formulas (25) and (26) that 119879119906119895has the

alternative representation

(119879119904

119906119895119891) (t) = int

R

119866119906119895(119909119895) 119891 (t + 119894119909119895e119895) 119889119909119895 x isin R

119889 (27)

We define the value of a kernel 119866u at x = (1199091 1199092 119909119889) by

119866u (x) = prod

119894isin119885119889

119866119906119894(119909119894) (28)

and introduce the operator

119879119904

u = 119879119904

1199061∘ sdot sdot sdot ∘ 119879

119904

119906119889 (29)

Consequently 119879119904

u is given by

(119879119904

u119891) (t) = intR119889

119866u (x) 119891 (t + x) 119889x t isin R119889 (30)

It is known from [20] that 119879119904

u119891 isin 119861119901

2119904u(R119889) We will exploit

the following properties of 119879119904

u119891 in the proof of Theorem 1

Lemma3 Let119891 isin U(119861r119901120579(R119889

)) 1 le 119901 and 120579 le infin For120590 gt 0define u isin R119889 with 119906119895 = 120590

119892(r)119903119895 for 119895 isin 119885119889 then one has

1003817100381710038171003817119891 minus 119879119904

u1198911003817100381710038171003817119871infin

le 119862 sdot 120590minus119892(r)+1119901

(31)

Proof When 119901 = infin the inequality was proved in [12] Bythe imbedding relationship

U (119861r119901120579

(R119889)) sub 119862 sdotU (119861

r1015840infin120579

(R119889)) (32)

where r1015840 = (1minus(1119901)(sum119889

119895=1(1119903119895)))r (see [20] formore details)

we can derive the corresponding inequalities for the case 1 le

119901 lt infin from that of 119901 = infin

Lemma 4 (see [22]) If 119891 isin 119871119901(R119889) 1 le 119901 le infin and u isin R119889

+

then

1003817100381710038171003817119879119904

u1198911003817100381710038171003817119871119901

le 21205721003817100381710038171003817119891

1003817100381710038171003817119871119901 (33)

where 120572 = sum119894isin119885119889119897119894

For 1 le 119901 lt infin let 119897119901(Z119889) be the Banach space of all infinite

bounded 119901-summable sequences y = 119910kkisinZ119889 such that thenorm

1003817100381710038171003817y1003817100381710038171003817119897119901

= ( sum

kisinZ119889|119910k|

119901)

1119901

(34)

is finite

Lemma 5 (see [10]) Let y = 119910k isin 119897119901(Z119889

) 1 lt 119901 lt infin Thenthe series

119871k (y t) = sum

kisinZ119889119910k sinc (k sdot t minus k) (35)

converges uniformly on R119889 to a function in 119861119901

k (R119889)

We also need the following bound for sinc seriessumkisinZ119889 |sinc(k sdot t minus k)|119902

Lemma 6 (see [11]) Let 119889 ge 1 119902 gt 1 k = (V1 V119889) andV119894 gt 1 119894 = 1 119889 Then for any t isin R119889

( sum

kisinZ119889|sinc (k sdot t minus k)|119902)

1119902

le (119902

119902 minus 1)

119889

(36)

For 119891 isin 119861119901

k (R119889) one has the following Marcinkiewicz-type

inequality

Lemma 7 (see [20 23]) Let 119891 isin 119861119901

k (R119889) 1 le 119901 lt infin Then

one has

(

119889

prod

119894=1

Vminus1119894

sum

kisinZ119889

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901 (37)

The next lemma presents a Marcinkiewicz-type inequal-ity for functions from Sobolev spaces

Abstract and Applied Analysis 5

Lemma 8 (see [10]) Let 119891 isin 119882119897

119901(R119889

) 1 le 119901 lt infin and 119897 ge 119889Then

(

119889

prod

119894=1

1

V119894sum

kisinZ119889

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

le 119862(1003817100381710038171003817119891

1003817100381710038171003817119871119901+

119889

sum

119894=1

1

V119894

10038171003817100381710038171003817100381710038171003817

120597119891

120597119909119894

10038171003817100381710038171003817100381710038171003817119871119901

+ sum

1le119894le119895le119889

1

V119894

1

V119895

1003817100381710038171003817100381710038171003817100381710038171003817

1205972119891

120597119909119894120597119909119895

1003817100381710038171003817100381710038171003817100381710038171003817119871119901

+ sdot sdot sdot +

119889

prod

119894=1

1

V119894

100381710038171003817100381710038171003817100381710038171003817

120597119889119891

1205971199091 sdot sdot sdot 120597119909119889

100381710038171003817100381710038171003817100381710038171003817119871119901

)

(38)

Lemma 9 (see [20 24]) Let l = (1198971 119897119889) isin N119889 r =

(1199031 119903119889) isin R119889

+ 1 le 119901 120579 le infin 1 minus sum

119889

119894=1119897119894119903119894 gt 0 and

r1015840 = (1 minus sum119889

119894=1119897119894119903119894)r For 119891 isin 119861

r119901120579(R119889

) it follows that thereexists a constant119862 depending on r r1015840 119901 and 120579 but independentof 119891 such that

10038171003817100381710038171003817120597|l|11989110038171003817100381710038171003817119861r1015840119901120579

le 1198621003817100381710038171003817119891

1003817100381710038171003817119861r119901120579

(39)

Proof of Theorem 1 It is known from Lemma 9 (letting 119897119894 = 1

for 119894 isin 119885119889) that the fact 119891 isin U(119861rp120579(R

119889)) with 119903119894 gt 119889 119894 isin 119885119889

implies 119891 isin 119882119889

119901(R119889

) therefore by Lemma 810038171003817100381710038171003817100381710038171003817

119891(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

1119901 (40)

And hence by Lemma 5

(119878k119891) (t) = sum

kisinZ119889119891(

kk) sinc (k sdot t minus k) (41)

converges uniformly on R119889Set u = k2119904 and 2119904 gt 119889 +max119903119894 119894 = 1 119889 So 119879

119904

u119891 isin

119861119901

k (R119889) as mentioned above By Theorem A we have 119879

119904

u119891 =

119878k(119879119904

u119891) Thus

(119879119904

u119891) (t) minus (119878k119891) (t)

= sum

kisinZ119889((119879

119904

u119891)(kk) minus 119891(

kk)) sinc (k sdot t minus k)

(42)

Using the triangle inequality we obtain1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 +

1003816100381610038161003816(119879119904

u119891) (t) minus (119878k119891) (t)1003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 +

1003816100381610038161003816119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

+1003816100381610038161003816119878k (119879

119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

(43)

By Lemma 31003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 le 119862120590

minus119892(r)+1119901 (44)

It is clear that1003816100381610038161003816119878kN (119879

119904

u119891 minus 119891) (t)1003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminuskisin119868119889N

(119879119904

u119891 minus 119891)(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(45)

Applying Holderrsquos inequality with exponent 1199010 we get1003816100381610038161003816119878kN (119879

119904

u119891 minus 119891) (t)1003816100381610038161003816

le ( sum

ksdottminuskisin119868119889N

10038161003816100381610038161003816100381610038161003816

(119879119904

u119891 minus 119891)(kk)

10038161003816100381610038161003816100381610038161003816

1199010

)

11199010

sdot ( sum

ksdottminuskisin119868119889N

|sinc(k sdot t minus k)|1199020)11199020

le 119862(

119889

prod

119894=1

2119873119894 + 1)

11199010

120590minus119892(r)+1119901

sdot 119901119889

0

(46)

where 11199010+11199020 = 1 and the second inequality follows from(44) and Lemma 6

Next we estimate |119878k(119879119904

u119891 minus 119891)(t) minus 119878kN(119879119904

u119891 minus 119891)(t)| ByHolderrsquos inequality

1003816100381610038161003816119878k (119879119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

le ( sum

ksdottminusknotin119868119889N

10038161003816100381610038161003816100381610038161003816

(119879119904

u119891 minus 119891)(kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

sdot ( sum

ksdottminusknotin119868119889N

100381610038161003816100381610038161003816sinc(k sdot t minus k)

100381610038161003816100381610038161003816

119902

)

1119902

(47)

where 1119901 + 1119902 = 1By Lemma 7 and Lemma 4 we obtain

10038171003817100381710038171003817100381710038171003817

(119879119904

u119891)(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

11199011003817100381710038171003817119879119904

u1198911003817100381710038171003817119871119901

le 11986212059011199011003817100381710038171003817119891

1003817100381710038171003817119871119901le 119862120590

1119901

(48)

It follows from (40) (48) and Minkowski inequality10038171003817100381710038171003817100381710038171003817

(119879119904

u119891 minus 119891)(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

1119901 (49)

Set ℎ(t) = (sumksdottminusknotin119868119889N|sinc(k sdot t minus k)|119902)1119902 Note that ℎ(t +

mk) = ℎ(t) for all t isin R119889 and m isin Z119889 Thus to give anupper estimate for ℎ(t) on R119889 we only need to bound it onprod

119889

119894=1[0 1V119894] Note that

k k notin 119868119889

N sub

119889

⋃

119894=1

k 119896119894 notin [minus119873119894 119873119894] (50)

6 Abstract and Applied Analysis

A straightforward computation shows that for 119905119894 isin [0 1V119894]

( sum

119896119894notin(minus119873119894 119873119894]

|sinc (V119894119905119894 minus 119896119894)|119902)

1119902

le (119862 sum

119896119894notin(minus119873119894 119873119894]

1

|119896|119902)

1119902

le (119862int

infin

119873119894

119905minus119902)

1119902

le 119862119873minus1119901

119894

(51)

Therefore

1003816100381610038161003816119878k (119879119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816 le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894)

(52)

It follows from (46) and (52) that

1003816100381610038161003816(119879119904

u119891 minus 119878k119891) (t)1003816100381610038161003816

le 119862((

119889

prod

119894=1

(2119873119894 + 1))

11199010

120590minus119892(r)+1119901

119901119889

0

+1205901119901

(

119889

sum

119894=1

119873minus1119901

119894))

(53)

We choose119873119894 = [(120590119892(r)

)119901

+ 1] and 1199010 = sum119889

119894=1ln(2119873119894 + 1) It is

easy to see that 1199010 gt 1 and (prod119889

119894=1(2119873119894 + 1))

11199010= 119890 A simple

computation gives

119901119889

0le 119862(ln

119889

prod

119894=1

119873119894)

119889

le 119862 ln119889120590 (54)

Note that119873119894 ge 120590119892(r)119901 Thus we have

119889

sum

119894=1

119873minus1119901

119894le 119862120590

minus119892(r) (55)

Collecting the above results we obtain

1003816100381610038161003816(119879119904

u119891) (t) minus (119878k119891) (t)1003816100381610038161003816 le 119862 sdot 120590

minus119892(r)+1119901ln119889120590 (56)

Combining (44) and (56) we prove the theorem

Proof of Theorem 2 By the triangle inequality we have

(119864kN119891) (t) le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816

+1003816100381610038161003816(119878k119891) (t) minus (119878kN119891) (t)

1003816100381610038161003816

(57)

By the arguments similar to those used in the proof ofTheorem 1 we obtain

1003816100381610038161003816(119878k119891) (t) minus (119878kN119891) (t)1003816100381610038161003816

le ( sum

ksdottminusknotin119868119889N

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

sdot ( sum

ksdottminusknotin119868119889N

100381610038161003816100381610038161003816sinc (k sdot t minus k)

100381610038161003816100381610038161003816

119902

)

1119902

le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894)

le 119862120590minus119892(r)+1119901ln119889120590

(58)

where we use119873119894 = [(120590119892(r)

)119901

+ 1] in the last inequalityCombiningTheorem 1 and (58) we complete the proof of

Theorem 2

3 The Measured Sampled Values Case

In practice the sampled values of a signal may not be exactlythe functional values and may have to be quantized Typicalerrors arising from these facts are jitter errors and amplitudeerrors Using the key idea of quasi-interpolation whichadopts integer translations of a basic function and integertranslations of a linear functional to approximate functionssee [8 25] and the references therein We may considersampled values that are the results of a linear functional andits integer translations acting on an undergoing signal [4 25]Such sampled values are called measured sampled valuesbecause they are closer to the truemeasurements taken fromasignalThe sampling series with themeasured sampled valuesis defined to be

(119878120582

k119891) (t) = sum

kisinZ119889120582119896119891(sdot +

kk) sinc (k sdot t minus k) (59)

where 120582 = 120582119896119896isinZ is any sequence of continuous linearfunctionals 1198620(R

119889) rarr C with 1198620(R

119889) being the set of all

continuous functions defined on R119889 and tending to zero atinfinity

Similar to the definition of (119878kN119891)(t) and (119864kN119891)(t) wehave the finite sum

(119878120582

kN119891) (t) = sum

kminusksdottisin119868119889N

120582119896119891(sdot +kk) sinc (k sdot t minus k) (60)

and the truncation error

(119864120582

kN119891) (t) =10038161003816100381610038161003816119891 (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816 (61)

To establish our theorems we need the error modulus

Ωk (119891 120582) = sup119896isinZ

10038161003816100381610038161003816100381610038161003816

120582119896119891(sdot +kk) minus 119891(

kk)

10038161003816100381610038161003816100381610038161003816

k gt 0 (62)

Abstract and Applied Analysis 7

WewriteΩ(119891 120582) forΩk(119891 120582) if no confusion arisesThe errormodulusΩ(119891 120582) provides a quantity for the quality of signalrsquosmeasured sampling values When the functionals in 120582 areconcrete we may get some reasonable estimates forΩ(119891 120582)

Sampling series with measured sampled values has beenstudied in [8] for bandlimited functions but without trunca-tionThe truncation errors are considered for functions fromLipschitz class with a decay condition in [13] Now we recalla typical result in [13]

Denote by Lip119871(1 119862(R119889

)) the set of all continuous func-tions 119891 satisfying

1003816100381610038161003816119891 (x) minus 119891 (y)1003816100381610038161003816 le 119871 (10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119909119889 minus 119910119889

1003816100381610038161003816) (63)

for all x = (1199091 119909119889) and y = (1199101 119910119889) isin R119889 SetkV = (1198961V 119896119889V) V sdot t = (V1199051 V119905119889) and tinfin =

max1199051 119905119889

Theorem D Let 119891 isin Lip119871(1 119862(R119889

)) satisfy the decay condi-tion

1003816100381610038161003816119891 (t)1003816100381610038161003816 le 119872119891tminus120574

infin tinfin ge 1 (64)

for some 0 lt 120574 le 1 Let 120582 = 120582119896 be any sequence of continuouslinear functionals For each V ge 119890

2 one has for the truncationerror at t isin R119889

10038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (t) minus sum

kisinZ119889kle119873120582119896119891(

kV) sinc (V sdot t minus k)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862Vminus1ln119889V

(65)

provided119873 = lfloor(V1+1120574minus1)2rfloor where lfloor119909rfloor is the smallest integerthat is greater or equal to a given 119909 isin R and Ω(119891 120582) le 1198880V

minus1

In [9] the author obtain the uniform bound of symmetrictruncation error for functions from isotropic Besov spacewith a similar decay condition Now we will provide theestimation for the truncation error

(119864120582

kN119891) (t) =10038161003816100381610038161003816119891 (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816

(66)

without any assumption about the decay of 119891 isin 119861r119901120579(R119889

)

Theorem 10 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define k as in Theorem B Let120582 = 120582119896 be any sequence of continuous linear functionals Fork one has

(119864120582

kN119891) (t) le 119862 sdot 120590minus119892(r)+1119901ln119889120590 (67)

provided 119873119894 = [(120590119892(r)

)119901

+ 1] and Ω(119891 120582) le 1198620120590minus119892(r)+1119901 for

some constant 1198620 gt 0

Proof By the triangle inequality we have

(119864120582

kN119891) (t)

le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 +10038161003816100381610038161003816(119878k119891) (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 + 1198681 + 1198682

(68)

where

1198681 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminusknotin119868119889N

119891(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198682 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminuskisin119868119889N

(119891(kk) minus 120582119896119891(sdot +

kk)) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)

Similar to (52) we have

1198681 le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894) (70)

Using Holderrsquos inequality we obtain

1198682 le ( sum

ksdottminuskisin119868119889N

10038161003816100381610038161003816100381610038161003816

119891 (kk) minus 120582119896119891(sdot +

kk)

10038161003816100381610038161003816100381610038161003816

1199010

)

11199010

sdot ( sum

ksdottminuskisin119868119889N

|sinc (k sdot t minus k) |1199020)11199020

le 119862(

119889

prod

119894=1

(2119873119894 + 1))

11199010

Ω(119891 120582) sdot 119901119889

0

(71)

where 11199010+11199020 = 1 Nowwe select the same119873119894 and1199010 as inthe proof ofTheorem 1 Similar to (55) we havesum119889

119894=1119873

minus1119901

119894le

119862120590minus119892(r) Thus

1198681 le 119862120590minus119892(r)+1119901

(72)

A simple computation gives (prod119889

119894=1(2119873119894 + 1))

11199010= 119890 and 1199010 le

119862 lnprod119889

119894=1119873119894 le 119862 ln120590 Notice that Ω(119891 120582) le 1198620120590

minus119892(r)+1119901Collecting these results we obtain

1198682 le 119862120590minus119892(r)+1119901ln119889120590 (73)

It follows fromTheorem 1 (72) and (73) that

(119864120582

kN119891) (t) le 119862120590minus119892(r)+1119901ln119889120590 (74)

which completes the proof

Finally we apply Theorem 10 to some practical examplesThe first one is that the measured sampled values are given byaverages of a function For119891 isin 1198620(R

119889)wedefine themodulus

of continuity

120596 (119891 120591) = suphle120591

1003817100381710038171003817119891 (sdot + h) minus 119891 (sdot)1003817100381710038171003817infin

(75)

where 120591may be any positive number

8 Abstract and Applied Analysis

Corollary 11 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define kN as in Theorem 10Suppose the sampled values 119891k of 119891 are obtained by the rule

119891k =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119891(t + kk)119889t (76)

where ℎ119895 are numbers satisfying 0 lt ℎ119895 le 120591 for all k isin Z119889

and 120591 gt 0 If 120591 lt 12 and 120596(119891 120591) le min119890119889(minus119892(r)+1119901)1198620120590

minus119892(r)+1119901 then

(119864120582

kN119891) (t) le 119862 sdot 120596(119891 120591)119892(r)minus1119901ln119889 1

120596 (119891 120591) (77)

Proof Let 120582 = 120582119896119896isinZ be the sequence of the linearfunctionals on 1198620(R

119889) and 119892 a continuous function on 119868

119889

h Define

120582119896119892 =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119892(t + kk)119889t (78)

Then 119891k = 120582119896119891(sdot + kk) Clearly |119891(t + kv) minus 119891(kv)| le

120596(119891 120591) Therefore

Ωk (119891 120582)

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816

120582119896119891(t + kk) minus 119891(

kk)

10038161003816100381610038161003816100381610038161003816

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

(119891(t + kk) minus 119891(

kk))119889t

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

120596 (119891 120591) 119889t10038161003816100381610038161003816100381610038161003816100381610038161003816

= 120596 (119891 120591)

(79)

Note that the function 119909 997891rarr 119909119892(r)minus1119901ln119889(1119909) is monotonely

increasing for 119909 isin (0 119890119889(minus119892(r)+1119901)

) The corollary followsfromTheorem 10

The second example is an estimate for the combination ofall four errors existing in sampling series the amplitude errorthe time-jitter error the truncation errors and the aliasingerrors We give some explanation for the amplitude error andthe time-jitter error

We assume the amplitude error results from quantizationwhich means the functional value 119891(119905) of a function 119891 atmoment 119905 is replaced by the nearest discrete value ormachinenumber 119891(119905) The quantization size is often known beforehand or can be chosen arbitrarily We may assume that thelocal error at any moment 119905 is bounded by a constant 120576 gt 0that is |119891(119905) minus 119891(119905)| le 120576 The time-jitter error arises if thesampled instances are notmet correctly butmight differ fromthe exact ones by 120591

1015840

119896 119896 isin Z we assume |1205911015840

119896| le 120591

1015840 for all 119896 andsome constant 1205911015840 gt 0 The combined error is defined to be

(119864119873119891) (119905) = 119891 (119905) minus sum

Vsdot119905minus119896isin[minus119873119873]

119891(119896

V+ 120591

1015840

119896) sinc (V119905 minus 119896)

(80)

Corollary 12 Let 119891 isin U(119861119903

119901120579(R)) 1 lt 119901 lt infin 1 le 120579 le infin

119903 gt 1 and V gt 119890 Then1003817100381710038171003817119864119873119891

1003817100381710038171003817infinle 119862Vminus119903+1119901 ln V (81)

provided119873 = [V119903119901+1] |119891(119905)minus119891(119905)| le 1198881Vminus119903+1119901 and120596(119891 120591

1015840) le

1198882Vminus119903+1119901 where 1198881 1198882 are positive constants and 1198881 + 1198882 le 1198620

Proof We define

120582119896 =

119891 (119896V + 1205911015840

119896)

119891 (119896V + 1205911015840

119896)120575 (sdot minus 120591

1015840

119896) (82)

where 120575 is the Dirac distribution Then 120582 = 120582119896119896isinZ is asequence of linear functional on1198620(R) It is clear that 120582119896119891(sdot+119896V) = 119891(119896V + 120591

1015840

119896) Then

10038161003816100381610038161003816100381610038161003816

120582119896119891(sdot +119896

V) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V+ 120591

1015840

119896)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le (1198881 + 1198882) Vminus119903+1119901

(83)

Thus Ω(119891 120582) le 1198620Vminus119903+1119901 By Theorem 10 we get the desired

result

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 10971251 11101220 and11271199) and the Program forNewCentury Excellent Talentsat University of China (NCET-10-0513)

References

[1] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 1948

[2] P L ButzerW Engels and U Scheben ldquoMagnitude of the trun-cation error in sam-pling expansions of bandlimited signalsrdquoIEEE Transactions on Acoustics Speech and Signal Processingvol 30 no 6 pp 906ndash912 1982

[3] A I ZayedAdvances in Shannonrsquos SamplingTheory CRC PressBoca Raton Fla USA 1993

[4] SDCasey andD FWalnut ldquoSystems of convolution equationsdeconvolution Shannon sampling and the wavelet and Gabortransformsrdquo SIAM Review vol 36 no 4 pp 537ndash577 1994

[5] P L Butzer G Schmeisser and R L Stens ldquoAn introductionto sampling analysisrdquo in Nonuniform Sampling Theory andPractice F Marvasti Ed pp 17ndash121 Kluwer Academic NewYork NY USA 2001

[6] S Smale and D-X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquo Bulletin of the AmericanMathematical Society vol 41 no 3 pp 279ndash305 2004

[7] S Smale and D-X Zhou ldquoShannon sampling II Connectionsto learning theoryrdquo Applied and Computational HarmonicAnalysis vol 19 no 3 pp 285ndash302 2005

Abstract and Applied Analysis 9

[8] P L Butzer and J Lei ldquoApproximation of signals usingmeasuredsampled values and error analysisrdquo Communications in AppliedAnalysis vol 4 no 2 pp 245ndash255 2000

[9] P X Ye ldquoError analysis for Shannon sampling series approxima-tionwithmeasured sampled valuesrdquoResearch Journal of AppliedSciences Engineering and Technology vol 5 no 3 pp 858ndash8642013

[10] J J Wang and G S Fang ldquoA multidimensional sampling theo-rem and an estimate of the aliasing errorrdquo Acta MathematicaeApplicatae Sinica vol 19 no 4 pp 481ndash488 1996

[11] X M Li ldquoUniform bounds for sampling expansionsrdquo Journal ofApproximation Theory vol 93 no 1 pp 100ndash113 1998

[12] L Jingfan and F Gensun ldquoOn uniform truncation error boundsand aliasing error for multidimensional sampling expansionrdquoSampling Theory in Signal and Image Processing vol 2 no 2pp 103ndash115 2003

[13] P L Butzer and J Lei ldquoErrors in truncated sampling serieswith measured sampled values for not-necessarily bandlimitedfunctionsrdquo Functiones et Approximatio vol 26 pp 25ndash39 1998

[14] H D Helms and J B Thomas ldquoTruncation error of sampling-theorem expansionsrdquo Proceedings of The IRE vol 50 no 2 pp179ndash184 1962

[15] D Jagerman ldquoBounds for truncation error of the samplingexpansionrdquo SIAM Journal on Applied Mathematics vol 14 no4 pp 714ndash723 1966

[16] C A Micchelli Y Xu and H Zhang ldquoOptimal learning ofbandlimited functions from localized samplingrdquo Journal ofComplexity vol 25 no 2 pp 85ndash114 2009

[17] A Ya Olenko and T K Pogany ldquoUniversal truncation errorupper bounds in sampling restorationrdquo Georgian MathematicalJournal vol 17 no 4 pp 765ndash786 2010

[18] P-X Ye and Z-H Song ldquoTruncation and aliasing errors forWhittaker-Kotelnikov-Shannon sampling expansionrdquo AppliedMathematics B vol 27 no 4 pp 412ndash418 2012

[19] P X Ye B H Sheng and X H Yuan ldquoOptimal order oftruncation and aliasing errors formulti-dimensional whittaker-shannon sampling expansionrdquo International Journal of Wirelessand Mobile Computing vol 5 no 4 pp 327ndash333 2012

[20] S M Nikolskii Approximation of Functions of Several Variablesand Imbedding Theorems Springer New York NY USA 1975

[21] Y Jiang and Y Liu ldquoAverage widths and optimal recovery ofmultivariate Besov classes in 119871119901(119877

119889)rdquo Journal of Approximation

Theory vol 102 no 1 pp 155ndash170 2000[22] C A Micchelli Y S Xu and P X Ye ldquoCucker-Smale learning

theory in Besov spacesrdquo in Advances in LearnIng TheoryMethods Models and Applications J Suykens G HorvathS Basu et al Eds pp 47ndash68 IOS Press Amsterdam TheNetherlands 2003

[23] R P Boas Jr Entire Functions Academic Press New York NYUSA 1954

[24] G Fang F J Hickernell and H Li ldquoApproximation onanisotropic Besov classes with mixed norms by standard infor-mationrdquo Journal of Complexity vol 21 no 3 pp 294ndash313 2005

[25] H G Burchard and J Lei ldquoCoordinate order of approxima-tion by functional-based approximation operatorsrdquo Journal ofApproximation Theory vol 82 no 2 pp 240ndash256 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

values taken at instances 119896V119896isinZ (cf [1]) In this case therepresentation of 119891 is given by

119891 (119905) = (119878V119891) (119905) =

+infin

sum

119896=minusinfin

119891(119896

V) sinc (V119905 minus 119896) (11)

where sinc (119905) = sin120587119905120587119905 119905 = 0 and sinc (0) = 1 Series (11)converges absolutely and uniformly on R

In [10] the authors establish multidimensional Shannonsampling theorem by extending (11) to the case 119891 isin 119861

119901

k (R119889)

1 lt 119901 lt infin and 119889 gt 1 They obtained the following theorem

TheoremA Let119891 isin 119861119901

k (R119889) 1 le 119901 lt infinThen for any t isin R119889

119891 (t) = (119878k119891) (t) = sum

kisinZ119889119891(

kk) sinc (k sdot t minus k) (12)

where sinc (t) = prod119889

119894=1sinc(119905119894) The series on the right-hand

side of (12) converges absolutely and uniformly on R119889

Shannonrsquos expansion requires us to know the exact valuesof a signal 119891 at infinitely many points and to sum an infiniteseries In practice only finitely many samples are availableand hence the symmetric truncation error

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (t) minus sum

|119896119889|le119873119889

sdot sdot sdot sum

|1198961|le1198731

119891(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(13)

has been widely studied under the assumption that119891 satisfiessome decay condition Among others in [11] the uniformtruncation error bounds are determined for 119891 isin 119861

2

V(R) witha decay condition In [12] the uniform bounds of truncationerror and aliasing error are derived for functions belongingto the Besov class 119861r

infin120579(R119889

) with the same decay condition asin [11] Since their results are the motivations of our workswe restate them as follows Throughout the paper we denotethe unit ball of the space 119861r

119901120579(R119889

) byU(119861r119901120579(R119889

))

Theorem B (see [12]) Let 119891 isin U(119861rinfin120579

(R119889)) 1 le 120579 le infin and

r isin R119889

+satisfy the decay condition inequality

1003816100381610038161003816119891 (t)1003816100381610038161003816 le119860

(1 + |t|2)120575 (14)

where 119860 gt 0 and 0 lt 120575 le 1 are constants and |t|2 = (1199052

1+

sdot sdot sdot + 1199052

119889)12 For 120590 gt 0 define the associated k = (V1 V119889) by

setting V119895 = 120590119892(r)119903119895 for 119895 isin 119885119889 If V119894 gt (12)119890

2120575 for 119894 isin 119885119889 then

1003816100381610038161003816119891 (t) minus (119878k119891) (t)1003816100381610038161003816 le 119862 sdot 120590

minus119892(r)ln119889120590 (15)

Theorem C (see [12]) Let 119891 isin U(119861rinfin120579

(R119889)) 1 le 120579 le

infin satisfy the decay condition (14) Then for any N =

(1198731 119873119889) isin N119889 with119873119894 gt (12)1198902120575 119894 = 1 119889 one has

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (t) minus sum

|119896119889|le119873119889

sdot sdot sdot sum

|1198961|le1198731

119891(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862(

119889

sum

119894=1

ln119873119894)

119889119889

sum

119894=1

119873minus119903119894(1+(119899120575)119903119894)

119894

(16)

Now we truncate the series on the right-hand side of (12)based on localized sampling That is if we want to estimate119891(t) we only sum over values of 119891 on a part of Z119889

k near tThus for any N isin N119889 we consider the finite sum

(119878kN119891) (t) = sum

kminusksdottisin119868119889N

119891(kk) sinc (k sdot t minus k) (17)

as an approximation to 119891(t) In this way we can derive theuniform bounds for the associated truncation error

(119864kN119891) (t) =1003816100381610038161003816119891 (t) minus (119878kN119891) (t)

1003816100381610038161003816 (18)

and aliasing error1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 (19)

without any assumption about the decay of 119891 isin U(119861r119901120579(R119889

))Our main result of this section is the following uniform

bound of the aliasing error1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 (20)

Theorem 1 Let 119891 isin U(119861r119901120579(R119889

)) with 1 lt 119901 lt infin 1 le 120579 le

infin and 119903119894 gt 119889 for 119894 isin 119885119889 For 120590 gt 119890 define k in the samemanner as in Theorem B then one has

1003816100381610038161003816119891 (t) minus (119878k119891) (t)1003816100381610038161003816 le 119862 sdot 120590

minus119892(r)+1119901ln119889120590 (21)

We firstly note that due to the localized sampling thefunction in Theorem 1 does not need to satisfy any decayassumption at infinity Next we make a comment on thebound 120590

minus119892(r)+1119901ln119889120590 It is known from the results of mean120590-dimension Kolmogorovwidths for Besov classU(119861

r119901120579(R119889

))

that

inf119892isin119861119901

k (R119889)

sup119891isinU(119861r

119901120579(R119889))

1003817100381710038171003817119891 minus 1198921003817100381710038171003817119871infin

ge 119862120590minus119892(r)+1119901

(22)

Thus the bound inTheorem 1 is optimal up to the logarithmicfactor ln119889120590 see [21] As a consequence ofTheorem 1 we showthat using truncated sampling series (17) we can still achievethis near optimal bound

4 Abstract and Applied Analysis

Theorem 2 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 and ras in Theorem 1 For 120590 gt 119890 define k as in Theorem B Then forN = (1198731 119873119889) isin N119889 with119873119894 = [(120590

119892(r))119901

+ 1] one has

(119864kN119891) (t) le 119862 sdot 120590minus119892(r)+1119901ln119889120590 (23)

To prove Theorem 1 we will choose an intermediatefunction which is a good approximation for both 119891 and 119878k119891Now we describe how to choose this function For moredetails one can see [21 22]

For any positive real number119906 gt 0 we define the function

119892119906 (119909) = 119860 119904(sinc120587minus1119906119909)

2119904 119909 isin R 2119904 gt 1 (24)

where the constant 119860 119904 is taken such that intR119892119906(119909)119889119909 = 1

Suppose 119906119895 gt 0 119895 isin 119885119889 For any 119891 isin 119861r119901120579

(R119889) set

(119879119904

119906119895119891) (t)

= intR

119892119906119895(119909119895) ((minus1)

119897119895+1 (Δ119897119895119909119895119891) (t) + 119891 (t)) 119889119909119895

= intR

119892119906119895(119909119895)

119897119895

sum

119894=1

119889119894119891 (1199051 119905119895minus1 119905119895 + 119894119909119895 119905119895+1 119905119889) 119889119909119895

(25)

where sum119897119895

119894=1119889119894 = 1

When 119905 isin R and 119895 isin 119885119889 we let

119866119906119895(119909) =

119897119895

sum

119894=1

119889119894

119894119892119906119895

(119909

119894) (26)

and observe from formulas (25) and (26) that 119879119906119895has the

alternative representation

(119879119904

119906119895119891) (t) = int

R

119866119906119895(119909119895) 119891 (t + 119894119909119895e119895) 119889119909119895 x isin R

119889 (27)

We define the value of a kernel 119866u at x = (1199091 1199092 119909119889) by

119866u (x) = prod

119894isin119885119889

119866119906119894(119909119894) (28)

and introduce the operator

119879119904

u = 119879119904

1199061∘ sdot sdot sdot ∘ 119879

119904

119906119889 (29)

Consequently 119879119904

u is given by

(119879119904

u119891) (t) = intR119889

119866u (x) 119891 (t + x) 119889x t isin R119889 (30)

It is known from [20] that 119879119904

u119891 isin 119861119901

2119904u(R119889) We will exploit

the following properties of 119879119904

u119891 in the proof of Theorem 1

Lemma3 Let119891 isin U(119861r119901120579(R119889

)) 1 le 119901 and 120579 le infin For120590 gt 0define u isin R119889 with 119906119895 = 120590

119892(r)119903119895 for 119895 isin 119885119889 then one has

1003817100381710038171003817119891 minus 119879119904

u1198911003817100381710038171003817119871infin

le 119862 sdot 120590minus119892(r)+1119901

(31)

Proof When 119901 = infin the inequality was proved in [12] Bythe imbedding relationship

U (119861r119901120579

(R119889)) sub 119862 sdotU (119861

r1015840infin120579

(R119889)) (32)

where r1015840 = (1minus(1119901)(sum119889

119895=1(1119903119895)))r (see [20] formore details)

we can derive the corresponding inequalities for the case 1 le

119901 lt infin from that of 119901 = infin

Lemma 4 (see [22]) If 119891 isin 119871119901(R119889) 1 le 119901 le infin and u isin R119889

+

then

1003817100381710038171003817119879119904

u1198911003817100381710038171003817119871119901

le 21205721003817100381710038171003817119891

1003817100381710038171003817119871119901 (33)

where 120572 = sum119894isin119885119889119897119894

For 1 le 119901 lt infin let 119897119901(Z119889) be the Banach space of all infinite

bounded 119901-summable sequences y = 119910kkisinZ119889 such that thenorm

1003817100381710038171003817y1003817100381710038171003817119897119901

= ( sum

kisinZ119889|119910k|

119901)

1119901

(34)

is finite

Lemma 5 (see [10]) Let y = 119910k isin 119897119901(Z119889

) 1 lt 119901 lt infin Thenthe series

119871k (y t) = sum

kisinZ119889119910k sinc (k sdot t minus k) (35)

converges uniformly on R119889 to a function in 119861119901

k (R119889)

We also need the following bound for sinc seriessumkisinZ119889 |sinc(k sdot t minus k)|119902

Lemma 6 (see [11]) Let 119889 ge 1 119902 gt 1 k = (V1 V119889) andV119894 gt 1 119894 = 1 119889 Then for any t isin R119889

( sum

kisinZ119889|sinc (k sdot t minus k)|119902)

1119902

le (119902

119902 minus 1)

119889

(36)

For 119891 isin 119861119901

k (R119889) one has the following Marcinkiewicz-type

inequality

Lemma 7 (see [20 23]) Let 119891 isin 119861119901

k (R119889) 1 le 119901 lt infin Then

one has

(

119889

prod

119894=1

Vminus1119894

sum

kisinZ119889

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901 (37)

The next lemma presents a Marcinkiewicz-type inequal-ity for functions from Sobolev spaces

Abstract and Applied Analysis 5

Lemma 8 (see [10]) Let 119891 isin 119882119897

119901(R119889

) 1 le 119901 lt infin and 119897 ge 119889Then

(

119889

prod

119894=1

1

V119894sum

kisinZ119889

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

le 119862(1003817100381710038171003817119891

1003817100381710038171003817119871119901+

119889

sum

119894=1

1

V119894

10038171003817100381710038171003817100381710038171003817

120597119891

120597119909119894

10038171003817100381710038171003817100381710038171003817119871119901

+ sum

1le119894le119895le119889

1

V119894

1

V119895

1003817100381710038171003817100381710038171003817100381710038171003817

1205972119891

120597119909119894120597119909119895

1003817100381710038171003817100381710038171003817100381710038171003817119871119901

+ sdot sdot sdot +

119889

prod

119894=1

1

V119894

100381710038171003817100381710038171003817100381710038171003817

120597119889119891

1205971199091 sdot sdot sdot 120597119909119889

100381710038171003817100381710038171003817100381710038171003817119871119901

)

(38)

Lemma 9 (see [20 24]) Let l = (1198971 119897119889) isin N119889 r =

(1199031 119903119889) isin R119889

+ 1 le 119901 120579 le infin 1 minus sum

119889

119894=1119897119894119903119894 gt 0 and

r1015840 = (1 minus sum119889

119894=1119897119894119903119894)r For 119891 isin 119861

r119901120579(R119889

) it follows that thereexists a constant119862 depending on r r1015840 119901 and 120579 but independentof 119891 such that

10038171003817100381710038171003817120597|l|11989110038171003817100381710038171003817119861r1015840119901120579

le 1198621003817100381710038171003817119891

1003817100381710038171003817119861r119901120579

(39)

Proof of Theorem 1 It is known from Lemma 9 (letting 119897119894 = 1

for 119894 isin 119885119889) that the fact 119891 isin U(119861rp120579(R

119889)) with 119903119894 gt 119889 119894 isin 119885119889

implies 119891 isin 119882119889

119901(R119889

) therefore by Lemma 810038171003817100381710038171003817100381710038171003817

119891(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

1119901 (40)

And hence by Lemma 5

(119878k119891) (t) = sum

kisinZ119889119891(

kk) sinc (k sdot t minus k) (41)

converges uniformly on R119889Set u = k2119904 and 2119904 gt 119889 +max119903119894 119894 = 1 119889 So 119879

119904

u119891 isin

119861119901

k (R119889) as mentioned above By Theorem A we have 119879

119904

u119891 =

119878k(119879119904

u119891) Thus

(119879119904

u119891) (t) minus (119878k119891) (t)

= sum

kisinZ119889((119879

119904

u119891)(kk) minus 119891(

kk)) sinc (k sdot t minus k)

(42)

Using the triangle inequality we obtain1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 +

1003816100381610038161003816(119879119904

u119891) (t) minus (119878k119891) (t)1003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 +

1003816100381610038161003816119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

+1003816100381610038161003816119878k (119879

119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

(43)

By Lemma 31003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 le 119862120590

minus119892(r)+1119901 (44)

It is clear that1003816100381610038161003816119878kN (119879

119904

u119891 minus 119891) (t)1003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminuskisin119868119889N

(119879119904

u119891 minus 119891)(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(45)

Applying Holderrsquos inequality with exponent 1199010 we get1003816100381610038161003816119878kN (119879

119904

u119891 minus 119891) (t)1003816100381610038161003816

le ( sum

ksdottminuskisin119868119889N

10038161003816100381610038161003816100381610038161003816

(119879119904

u119891 minus 119891)(kk)

10038161003816100381610038161003816100381610038161003816

1199010

)

11199010

sdot ( sum

ksdottminuskisin119868119889N

|sinc(k sdot t minus k)|1199020)11199020

le 119862(

119889

prod

119894=1

2119873119894 + 1)

11199010

120590minus119892(r)+1119901

sdot 119901119889

0

(46)

where 11199010+11199020 = 1 and the second inequality follows from(44) and Lemma 6

Next we estimate |119878k(119879119904

u119891 minus 119891)(t) minus 119878kN(119879119904

u119891 minus 119891)(t)| ByHolderrsquos inequality

1003816100381610038161003816119878k (119879119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

le ( sum

ksdottminusknotin119868119889N

10038161003816100381610038161003816100381610038161003816

(119879119904

u119891 minus 119891)(kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

sdot ( sum

ksdottminusknotin119868119889N

100381610038161003816100381610038161003816sinc(k sdot t minus k)

100381610038161003816100381610038161003816

119902

)

1119902

(47)

where 1119901 + 1119902 = 1By Lemma 7 and Lemma 4 we obtain

10038171003817100381710038171003817100381710038171003817

(119879119904

u119891)(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

11199011003817100381710038171003817119879119904

u1198911003817100381710038171003817119871119901

le 11986212059011199011003817100381710038171003817119891

1003817100381710038171003817119871119901le 119862120590

1119901

(48)

It follows from (40) (48) and Minkowski inequality10038171003817100381710038171003817100381710038171003817

(119879119904

u119891 minus 119891)(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

1119901 (49)

Set ℎ(t) = (sumksdottminusknotin119868119889N|sinc(k sdot t minus k)|119902)1119902 Note that ℎ(t +

mk) = ℎ(t) for all t isin R119889 and m isin Z119889 Thus to give anupper estimate for ℎ(t) on R119889 we only need to bound it onprod

119889

119894=1[0 1V119894] Note that

k k notin 119868119889

N sub

119889

⋃

119894=1

k 119896119894 notin [minus119873119894 119873119894] (50)

6 Abstract and Applied Analysis

A straightforward computation shows that for 119905119894 isin [0 1V119894]

( sum

119896119894notin(minus119873119894 119873119894]

|sinc (V119894119905119894 minus 119896119894)|119902)

1119902

le (119862 sum

119896119894notin(minus119873119894 119873119894]

1

|119896|119902)

1119902

le (119862int

infin

119873119894

119905minus119902)

1119902

le 119862119873minus1119901

119894

(51)

Therefore

1003816100381610038161003816119878k (119879119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816 le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894)

(52)

It follows from (46) and (52) that

1003816100381610038161003816(119879119904

u119891 minus 119878k119891) (t)1003816100381610038161003816

le 119862((

119889

prod

119894=1

(2119873119894 + 1))

11199010

120590minus119892(r)+1119901

119901119889

0

+1205901119901

(

119889

sum

119894=1

119873minus1119901

119894))

(53)

We choose119873119894 = [(120590119892(r)

)119901

+ 1] and 1199010 = sum119889

119894=1ln(2119873119894 + 1) It is

easy to see that 1199010 gt 1 and (prod119889

119894=1(2119873119894 + 1))

11199010= 119890 A simple

computation gives

119901119889

0le 119862(ln

119889

prod

119894=1

119873119894)

119889

le 119862 ln119889120590 (54)

Note that119873119894 ge 120590119892(r)119901 Thus we have

119889

sum

119894=1

119873minus1119901

119894le 119862120590

minus119892(r) (55)

Collecting the above results we obtain

1003816100381610038161003816(119879119904

u119891) (t) minus (119878k119891) (t)1003816100381610038161003816 le 119862 sdot 120590

minus119892(r)+1119901ln119889120590 (56)

Combining (44) and (56) we prove the theorem

Proof of Theorem 2 By the triangle inequality we have

(119864kN119891) (t) le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816

+1003816100381610038161003816(119878k119891) (t) minus (119878kN119891) (t)

1003816100381610038161003816

(57)

By the arguments similar to those used in the proof ofTheorem 1 we obtain

1003816100381610038161003816(119878k119891) (t) minus (119878kN119891) (t)1003816100381610038161003816

le ( sum

ksdottminusknotin119868119889N

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

sdot ( sum

ksdottminusknotin119868119889N

100381610038161003816100381610038161003816sinc (k sdot t minus k)

100381610038161003816100381610038161003816

119902

)

1119902

le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894)

le 119862120590minus119892(r)+1119901ln119889120590

(58)

where we use119873119894 = [(120590119892(r)

)119901

+ 1] in the last inequalityCombiningTheorem 1 and (58) we complete the proof of

Theorem 2

3 The Measured Sampled Values Case

In practice the sampled values of a signal may not be exactlythe functional values and may have to be quantized Typicalerrors arising from these facts are jitter errors and amplitudeerrors Using the key idea of quasi-interpolation whichadopts integer translations of a basic function and integertranslations of a linear functional to approximate functionssee [8 25] and the references therein We may considersampled values that are the results of a linear functional andits integer translations acting on an undergoing signal [4 25]Such sampled values are called measured sampled valuesbecause they are closer to the truemeasurements taken fromasignalThe sampling series with themeasured sampled valuesis defined to be

(119878120582

k119891) (t) = sum

kisinZ119889120582119896119891(sdot +

kk) sinc (k sdot t minus k) (59)

where 120582 = 120582119896119896isinZ is any sequence of continuous linearfunctionals 1198620(R

119889) rarr C with 1198620(R

119889) being the set of all

continuous functions defined on R119889 and tending to zero atinfinity

Similar to the definition of (119878kN119891)(t) and (119864kN119891)(t) wehave the finite sum

(119878120582

kN119891) (t) = sum

kminusksdottisin119868119889N

120582119896119891(sdot +kk) sinc (k sdot t minus k) (60)

and the truncation error

(119864120582

kN119891) (t) =10038161003816100381610038161003816119891 (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816 (61)

To establish our theorems we need the error modulus

Ωk (119891 120582) = sup119896isinZ

10038161003816100381610038161003816100381610038161003816

120582119896119891(sdot +kk) minus 119891(

kk)

10038161003816100381610038161003816100381610038161003816

k gt 0 (62)

Abstract and Applied Analysis 7

WewriteΩ(119891 120582) forΩk(119891 120582) if no confusion arisesThe errormodulusΩ(119891 120582) provides a quantity for the quality of signalrsquosmeasured sampling values When the functionals in 120582 areconcrete we may get some reasonable estimates forΩ(119891 120582)

Sampling series with measured sampled values has beenstudied in [8] for bandlimited functions but without trunca-tionThe truncation errors are considered for functions fromLipschitz class with a decay condition in [13] Now we recalla typical result in [13]

Denote by Lip119871(1 119862(R119889

)) the set of all continuous func-tions 119891 satisfying

1003816100381610038161003816119891 (x) minus 119891 (y)1003816100381610038161003816 le 119871 (10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119909119889 minus 119910119889

1003816100381610038161003816) (63)

for all x = (1199091 119909119889) and y = (1199101 119910119889) isin R119889 SetkV = (1198961V 119896119889V) V sdot t = (V1199051 V119905119889) and tinfin =

max1199051 119905119889

Theorem D Let 119891 isin Lip119871(1 119862(R119889

)) satisfy the decay condi-tion

1003816100381610038161003816119891 (t)1003816100381610038161003816 le 119872119891tminus120574

infin tinfin ge 1 (64)

for some 0 lt 120574 le 1 Let 120582 = 120582119896 be any sequence of continuouslinear functionals For each V ge 119890

2 one has for the truncationerror at t isin R119889

10038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (t) minus sum

kisinZ119889kle119873120582119896119891(

kV) sinc (V sdot t minus k)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862Vminus1ln119889V

(65)

provided119873 = lfloor(V1+1120574minus1)2rfloor where lfloor119909rfloor is the smallest integerthat is greater or equal to a given 119909 isin R and Ω(119891 120582) le 1198880V

minus1

In [9] the author obtain the uniform bound of symmetrictruncation error for functions from isotropic Besov spacewith a similar decay condition Now we will provide theestimation for the truncation error

(119864120582

kN119891) (t) =10038161003816100381610038161003816119891 (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816

(66)

without any assumption about the decay of 119891 isin 119861r119901120579(R119889

)

Theorem 10 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define k as in Theorem B Let120582 = 120582119896 be any sequence of continuous linear functionals Fork one has

(119864120582

kN119891) (t) le 119862 sdot 120590minus119892(r)+1119901ln119889120590 (67)

provided 119873119894 = [(120590119892(r)

)119901

+ 1] and Ω(119891 120582) le 1198620120590minus119892(r)+1119901 for

some constant 1198620 gt 0

Proof By the triangle inequality we have

(119864120582

kN119891) (t)

le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 +10038161003816100381610038161003816(119878k119891) (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 + 1198681 + 1198682

(68)

where

1198681 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminusknotin119868119889N

119891(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198682 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminuskisin119868119889N

(119891(kk) minus 120582119896119891(sdot +

kk)) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)

Similar to (52) we have

1198681 le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894) (70)

Using Holderrsquos inequality we obtain

1198682 le ( sum

ksdottminuskisin119868119889N

10038161003816100381610038161003816100381610038161003816

119891 (kk) minus 120582119896119891(sdot +

kk)

10038161003816100381610038161003816100381610038161003816

1199010

)

11199010

sdot ( sum

ksdottminuskisin119868119889N

|sinc (k sdot t minus k) |1199020)11199020

le 119862(

119889

prod

119894=1

(2119873119894 + 1))

11199010

Ω(119891 120582) sdot 119901119889

0

(71)

where 11199010+11199020 = 1 Nowwe select the same119873119894 and1199010 as inthe proof ofTheorem 1 Similar to (55) we havesum119889

119894=1119873

minus1119901

119894le

119862120590minus119892(r) Thus

1198681 le 119862120590minus119892(r)+1119901

(72)

A simple computation gives (prod119889

119894=1(2119873119894 + 1))

11199010= 119890 and 1199010 le

119862 lnprod119889

119894=1119873119894 le 119862 ln120590 Notice that Ω(119891 120582) le 1198620120590

minus119892(r)+1119901Collecting these results we obtain

1198682 le 119862120590minus119892(r)+1119901ln119889120590 (73)

It follows fromTheorem 1 (72) and (73) that

(119864120582

kN119891) (t) le 119862120590minus119892(r)+1119901ln119889120590 (74)

which completes the proof

Finally we apply Theorem 10 to some practical examplesThe first one is that the measured sampled values are given byaverages of a function For119891 isin 1198620(R

119889)wedefine themodulus

of continuity

120596 (119891 120591) = suphle120591

1003817100381710038171003817119891 (sdot + h) minus 119891 (sdot)1003817100381710038171003817infin

(75)

where 120591may be any positive number

8 Abstract and Applied Analysis

Corollary 11 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define kN as in Theorem 10Suppose the sampled values 119891k of 119891 are obtained by the rule

119891k =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119891(t + kk)119889t (76)

where ℎ119895 are numbers satisfying 0 lt ℎ119895 le 120591 for all k isin Z119889

and 120591 gt 0 If 120591 lt 12 and 120596(119891 120591) le min119890119889(minus119892(r)+1119901)1198620120590

minus119892(r)+1119901 then

(119864120582

kN119891) (t) le 119862 sdot 120596(119891 120591)119892(r)minus1119901ln119889 1

120596 (119891 120591) (77)

Proof Let 120582 = 120582119896119896isinZ be the sequence of the linearfunctionals on 1198620(R

119889) and 119892 a continuous function on 119868

119889

h Define

120582119896119892 =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119892(t + kk)119889t (78)

Then 119891k = 120582119896119891(sdot + kk) Clearly |119891(t + kv) minus 119891(kv)| le

120596(119891 120591) Therefore

Ωk (119891 120582)

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816

120582119896119891(t + kk) minus 119891(

kk)

10038161003816100381610038161003816100381610038161003816

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

(119891(t + kk) minus 119891(

kk))119889t

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

120596 (119891 120591) 119889t10038161003816100381610038161003816100381610038161003816100381610038161003816

= 120596 (119891 120591)

(79)

Note that the function 119909 997891rarr 119909119892(r)minus1119901ln119889(1119909) is monotonely

increasing for 119909 isin (0 119890119889(minus119892(r)+1119901)

) The corollary followsfromTheorem 10

The second example is an estimate for the combination ofall four errors existing in sampling series the amplitude errorthe time-jitter error the truncation errors and the aliasingerrors We give some explanation for the amplitude error andthe time-jitter error

We assume the amplitude error results from quantizationwhich means the functional value 119891(119905) of a function 119891 atmoment 119905 is replaced by the nearest discrete value ormachinenumber 119891(119905) The quantization size is often known beforehand or can be chosen arbitrarily We may assume that thelocal error at any moment 119905 is bounded by a constant 120576 gt 0that is |119891(119905) minus 119891(119905)| le 120576 The time-jitter error arises if thesampled instances are notmet correctly butmight differ fromthe exact ones by 120591

1015840

119896 119896 isin Z we assume |1205911015840

119896| le 120591

1015840 for all 119896 andsome constant 1205911015840 gt 0 The combined error is defined to be

(119864119873119891) (119905) = 119891 (119905) minus sum

Vsdot119905minus119896isin[minus119873119873]

119891(119896

V+ 120591

1015840

119896) sinc (V119905 minus 119896)

(80)

Corollary 12 Let 119891 isin U(119861119903

119901120579(R)) 1 lt 119901 lt infin 1 le 120579 le infin

119903 gt 1 and V gt 119890 Then1003817100381710038171003817119864119873119891

1003817100381710038171003817infinle 119862Vminus119903+1119901 ln V (81)

provided119873 = [V119903119901+1] |119891(119905)minus119891(119905)| le 1198881Vminus119903+1119901 and120596(119891 120591

1015840) le

1198882Vminus119903+1119901 where 1198881 1198882 are positive constants and 1198881 + 1198882 le 1198620

Proof We define

120582119896 =

119891 (119896V + 1205911015840

119896)

119891 (119896V + 1205911015840

119896)120575 (sdot minus 120591

1015840

119896) (82)

where 120575 is the Dirac distribution Then 120582 = 120582119896119896isinZ is asequence of linear functional on1198620(R) It is clear that 120582119896119891(sdot+119896V) = 119891(119896V + 120591

1015840

119896) Then

10038161003816100381610038161003816100381610038161003816

120582119896119891(sdot +119896

V) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V+ 120591

1015840

119896)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le (1198881 + 1198882) Vminus119903+1119901

(83)

Thus Ω(119891 120582) le 1198620Vminus119903+1119901 By Theorem 10 we get the desired

result

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 10971251 11101220 and11271199) and the Program forNewCentury Excellent Talentsat University of China (NCET-10-0513)

References

[1] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 1948

[2] P L ButzerW Engels and U Scheben ldquoMagnitude of the trun-cation error in sam-pling expansions of bandlimited signalsrdquoIEEE Transactions on Acoustics Speech and Signal Processingvol 30 no 6 pp 906ndash912 1982

[3] A I ZayedAdvances in Shannonrsquos SamplingTheory CRC PressBoca Raton Fla USA 1993

[4] SDCasey andD FWalnut ldquoSystems of convolution equationsdeconvolution Shannon sampling and the wavelet and Gabortransformsrdquo SIAM Review vol 36 no 4 pp 537ndash577 1994

[5] P L Butzer G Schmeisser and R L Stens ldquoAn introductionto sampling analysisrdquo in Nonuniform Sampling Theory andPractice F Marvasti Ed pp 17ndash121 Kluwer Academic NewYork NY USA 2001

[6] S Smale and D-X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquo Bulletin of the AmericanMathematical Society vol 41 no 3 pp 279ndash305 2004

[7] S Smale and D-X Zhou ldquoShannon sampling II Connectionsto learning theoryrdquo Applied and Computational HarmonicAnalysis vol 19 no 3 pp 285ndash302 2005

Abstract and Applied Analysis 9

[8] P L Butzer and J Lei ldquoApproximation of signals usingmeasuredsampled values and error analysisrdquo Communications in AppliedAnalysis vol 4 no 2 pp 245ndash255 2000

[9] P X Ye ldquoError analysis for Shannon sampling series approxima-tionwithmeasured sampled valuesrdquoResearch Journal of AppliedSciences Engineering and Technology vol 5 no 3 pp 858ndash8642013

[10] J J Wang and G S Fang ldquoA multidimensional sampling theo-rem and an estimate of the aliasing errorrdquo Acta MathematicaeApplicatae Sinica vol 19 no 4 pp 481ndash488 1996

[11] X M Li ldquoUniform bounds for sampling expansionsrdquo Journal ofApproximation Theory vol 93 no 1 pp 100ndash113 1998

[12] L Jingfan and F Gensun ldquoOn uniform truncation error boundsand aliasing error for multidimensional sampling expansionrdquoSampling Theory in Signal and Image Processing vol 2 no 2pp 103ndash115 2003

[13] P L Butzer and J Lei ldquoErrors in truncated sampling serieswith measured sampled values for not-necessarily bandlimitedfunctionsrdquo Functiones et Approximatio vol 26 pp 25ndash39 1998

[14] H D Helms and J B Thomas ldquoTruncation error of sampling-theorem expansionsrdquo Proceedings of The IRE vol 50 no 2 pp179ndash184 1962

[15] D Jagerman ldquoBounds for truncation error of the samplingexpansionrdquo SIAM Journal on Applied Mathematics vol 14 no4 pp 714ndash723 1966

[16] C A Micchelli Y Xu and H Zhang ldquoOptimal learning ofbandlimited functions from localized samplingrdquo Journal ofComplexity vol 25 no 2 pp 85ndash114 2009

[17] A Ya Olenko and T K Pogany ldquoUniversal truncation errorupper bounds in sampling restorationrdquo Georgian MathematicalJournal vol 17 no 4 pp 765ndash786 2010

[18] P-X Ye and Z-H Song ldquoTruncation and aliasing errors forWhittaker-Kotelnikov-Shannon sampling expansionrdquo AppliedMathematics B vol 27 no 4 pp 412ndash418 2012

[19] P X Ye B H Sheng and X H Yuan ldquoOptimal order oftruncation and aliasing errors formulti-dimensional whittaker-shannon sampling expansionrdquo International Journal of Wirelessand Mobile Computing vol 5 no 4 pp 327ndash333 2012

[20] S M Nikolskii Approximation of Functions of Several Variablesand Imbedding Theorems Springer New York NY USA 1975

[21] Y Jiang and Y Liu ldquoAverage widths and optimal recovery ofmultivariate Besov classes in 119871119901(119877

119889)rdquo Journal of Approximation

Theory vol 102 no 1 pp 155ndash170 2000[22] C A Micchelli Y S Xu and P X Ye ldquoCucker-Smale learning

theory in Besov spacesrdquo in Advances in LearnIng TheoryMethods Models and Applications J Suykens G HorvathS Basu et al Eds pp 47ndash68 IOS Press Amsterdam TheNetherlands 2003

[23] R P Boas Jr Entire Functions Academic Press New York NYUSA 1954

[24] G Fang F J Hickernell and H Li ldquoApproximation onanisotropic Besov classes with mixed norms by standard infor-mationrdquo Journal of Complexity vol 21 no 3 pp 294ndash313 2005

[25] H G Burchard and J Lei ldquoCoordinate order of approxima-tion by functional-based approximation operatorsrdquo Journal ofApproximation Theory vol 82 no 2 pp 240ndash256 1995

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Theorem 2 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 and ras in Theorem 1 For 120590 gt 119890 define k as in Theorem B Then forN = (1198731 119873119889) isin N119889 with119873119894 = [(120590

119892(r))119901

+ 1] one has

(119864kN119891) (t) le 119862 sdot 120590minus119892(r)+1119901ln119889120590 (23)

To prove Theorem 1 we will choose an intermediatefunction which is a good approximation for both 119891 and 119878k119891Now we describe how to choose this function For moredetails one can see [21 22]

For any positive real number119906 gt 0 we define the function

119892119906 (119909) = 119860 119904(sinc120587minus1119906119909)

2119904 119909 isin R 2119904 gt 1 (24)

where the constant 119860 119904 is taken such that intR119892119906(119909)119889119909 = 1

Suppose 119906119895 gt 0 119895 isin 119885119889 For any 119891 isin 119861r119901120579

(R119889) set

(119879119904

119906119895119891) (t)

= intR

119892119906119895(119909119895) ((minus1)

119897119895+1 (Δ119897119895119909119895119891) (t) + 119891 (t)) 119889119909119895

= intR

119892119906119895(119909119895)

119897119895

sum

119894=1

119889119894119891 (1199051 119905119895minus1 119905119895 + 119894119909119895 119905119895+1 119905119889) 119889119909119895

(25)

where sum119897119895

119894=1119889119894 = 1

When 119905 isin R and 119895 isin 119885119889 we let

119866119906119895(119909) =

119897119895

sum

119894=1

119889119894

119894119892119906119895

(119909

119894) (26)

and observe from formulas (25) and (26) that 119879119906119895has the

alternative representation

(119879119904

119906119895119891) (t) = int

R

119866119906119895(119909119895) 119891 (t + 119894119909119895e119895) 119889119909119895 x isin R

119889 (27)

We define the value of a kernel 119866u at x = (1199091 1199092 119909119889) by

119866u (x) = prod

119894isin119885119889

119866119906119894(119909119894) (28)

and introduce the operator

119879119904

u = 119879119904

1199061∘ sdot sdot sdot ∘ 119879

119904

119906119889 (29)

Consequently 119879119904

u is given by

(119879119904

u119891) (t) = intR119889

119866u (x) 119891 (t + x) 119889x t isin R119889 (30)

It is known from [20] that 119879119904

u119891 isin 119861119901

2119904u(R119889) We will exploit

the following properties of 119879119904

u119891 in the proof of Theorem 1

Lemma3 Let119891 isin U(119861r119901120579(R119889

)) 1 le 119901 and 120579 le infin For120590 gt 0define u isin R119889 with 119906119895 = 120590

119892(r)119903119895 for 119895 isin 119885119889 then one has

1003817100381710038171003817119891 minus 119879119904

u1198911003817100381710038171003817119871infin

le 119862 sdot 120590minus119892(r)+1119901

(31)

Proof When 119901 = infin the inequality was proved in [12] Bythe imbedding relationship

U (119861r119901120579

(R119889)) sub 119862 sdotU (119861

r1015840infin120579

(R119889)) (32)

where r1015840 = (1minus(1119901)(sum119889

119895=1(1119903119895)))r (see [20] formore details)

we can derive the corresponding inequalities for the case 1 le

119901 lt infin from that of 119901 = infin

Lemma 4 (see [22]) If 119891 isin 119871119901(R119889) 1 le 119901 le infin and u isin R119889

+

then

1003817100381710038171003817119879119904

u1198911003817100381710038171003817119871119901

le 21205721003817100381710038171003817119891

1003817100381710038171003817119871119901 (33)

where 120572 = sum119894isin119885119889119897119894

For 1 le 119901 lt infin let 119897119901(Z119889) be the Banach space of all infinite

bounded 119901-summable sequences y = 119910kkisinZ119889 such that thenorm

1003817100381710038171003817y1003817100381710038171003817119897119901

= ( sum

kisinZ119889|119910k|

119901)

1119901

(34)

is finite

Lemma 5 (see [10]) Let y = 119910k isin 119897119901(Z119889

) 1 lt 119901 lt infin Thenthe series

119871k (y t) = sum

kisinZ119889119910k sinc (k sdot t minus k) (35)

converges uniformly on R119889 to a function in 119861119901

k (R119889)

We also need the following bound for sinc seriessumkisinZ119889 |sinc(k sdot t minus k)|119902

Lemma 6 (see [11]) Let 119889 ge 1 119902 gt 1 k = (V1 V119889) andV119894 gt 1 119894 = 1 119889 Then for any t isin R119889

( sum

kisinZ119889|sinc (k sdot t minus k)|119902)

1119902

le (119902

119902 minus 1)

119889

(36)

For 119891 isin 119861119901

k (R119889) one has the following Marcinkiewicz-type

inequality

Lemma 7 (see [20 23]) Let 119891 isin 119861119901

k (R119889) 1 le 119901 lt infin Then

one has

(

119889

prod

119894=1

Vminus1119894

sum

kisinZ119889

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901 (37)

The next lemma presents a Marcinkiewicz-type inequal-ity for functions from Sobolev spaces

Abstract and Applied Analysis 5

Lemma 8 (see [10]) Let 119891 isin 119882119897

119901(R119889

) 1 le 119901 lt infin and 119897 ge 119889Then

(

119889

prod

119894=1

1

V119894sum

kisinZ119889

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

le 119862(1003817100381710038171003817119891

1003817100381710038171003817119871119901+

119889

sum

119894=1

1

V119894

10038171003817100381710038171003817100381710038171003817

120597119891

120597119909119894

10038171003817100381710038171003817100381710038171003817119871119901

+ sum

1le119894le119895le119889

1

V119894

1

V119895

1003817100381710038171003817100381710038171003817100381710038171003817

1205972119891

120597119909119894120597119909119895

1003817100381710038171003817100381710038171003817100381710038171003817119871119901

+ sdot sdot sdot +

119889

prod

119894=1

1

V119894

100381710038171003817100381710038171003817100381710038171003817

120597119889119891

1205971199091 sdot sdot sdot 120597119909119889

100381710038171003817100381710038171003817100381710038171003817119871119901

)

(38)

Lemma 9 (see [20 24]) Let l = (1198971 119897119889) isin N119889 r =

(1199031 119903119889) isin R119889

+ 1 le 119901 120579 le infin 1 minus sum

119889

119894=1119897119894119903119894 gt 0 and

r1015840 = (1 minus sum119889

119894=1119897119894119903119894)r For 119891 isin 119861

r119901120579(R119889

) it follows that thereexists a constant119862 depending on r r1015840 119901 and 120579 but independentof 119891 such that

10038171003817100381710038171003817120597|l|11989110038171003817100381710038171003817119861r1015840119901120579

le 1198621003817100381710038171003817119891

1003817100381710038171003817119861r119901120579

(39)

Proof of Theorem 1 It is known from Lemma 9 (letting 119897119894 = 1

for 119894 isin 119885119889) that the fact 119891 isin U(119861rp120579(R

119889)) with 119903119894 gt 119889 119894 isin 119885119889

implies 119891 isin 119882119889

119901(R119889

) therefore by Lemma 810038171003817100381710038171003817100381710038171003817

119891(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

1119901 (40)

And hence by Lemma 5

(119878k119891) (t) = sum

kisinZ119889119891(

kk) sinc (k sdot t minus k) (41)

converges uniformly on R119889Set u = k2119904 and 2119904 gt 119889 +max119903119894 119894 = 1 119889 So 119879

119904

u119891 isin

119861119901

k (R119889) as mentioned above By Theorem A we have 119879

119904

u119891 =

119878k(119879119904

u119891) Thus

(119879119904

u119891) (t) minus (119878k119891) (t)

= sum

kisinZ119889((119879

119904

u119891)(kk) minus 119891(

kk)) sinc (k sdot t minus k)

(42)

Using the triangle inequality we obtain1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 +

1003816100381610038161003816(119879119904

u119891) (t) minus (119878k119891) (t)1003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 +

1003816100381610038161003816119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

+1003816100381610038161003816119878k (119879

119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

(43)

By Lemma 31003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 le 119862120590

minus119892(r)+1119901 (44)

It is clear that1003816100381610038161003816119878kN (119879

119904

u119891 minus 119891) (t)1003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminuskisin119868119889N

(119879119904

u119891 minus 119891)(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(45)

Applying Holderrsquos inequality with exponent 1199010 we get1003816100381610038161003816119878kN (119879

119904

u119891 minus 119891) (t)1003816100381610038161003816

le ( sum

ksdottminuskisin119868119889N

10038161003816100381610038161003816100381610038161003816

(119879119904

u119891 minus 119891)(kk)

10038161003816100381610038161003816100381610038161003816

1199010

)

11199010

sdot ( sum

ksdottminuskisin119868119889N

|sinc(k sdot t minus k)|1199020)11199020

le 119862(

119889

prod

119894=1

2119873119894 + 1)

11199010

120590minus119892(r)+1119901

sdot 119901119889

0

(46)

where 11199010+11199020 = 1 and the second inequality follows from(44) and Lemma 6

Next we estimate |119878k(119879119904

u119891 minus 119891)(t) minus 119878kN(119879119904

u119891 minus 119891)(t)| ByHolderrsquos inequality

1003816100381610038161003816119878k (119879119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

le ( sum

ksdottminusknotin119868119889N

10038161003816100381610038161003816100381610038161003816

(119879119904

u119891 minus 119891)(kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

sdot ( sum

ksdottminusknotin119868119889N

100381610038161003816100381610038161003816sinc(k sdot t minus k)

100381610038161003816100381610038161003816

119902

)

1119902

(47)

where 1119901 + 1119902 = 1By Lemma 7 and Lemma 4 we obtain

10038171003817100381710038171003817100381710038171003817

(119879119904

u119891)(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

11199011003817100381710038171003817119879119904

u1198911003817100381710038171003817119871119901

le 11986212059011199011003817100381710038171003817119891

1003817100381710038171003817119871119901le 119862120590

1119901

(48)

It follows from (40) (48) and Minkowski inequality10038171003817100381710038171003817100381710038171003817

(119879119904

u119891 minus 119891)(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

1119901 (49)

Set ℎ(t) = (sumksdottminusknotin119868119889N|sinc(k sdot t minus k)|119902)1119902 Note that ℎ(t +

mk) = ℎ(t) for all t isin R119889 and m isin Z119889 Thus to give anupper estimate for ℎ(t) on R119889 we only need to bound it onprod

119889

119894=1[0 1V119894] Note that

k k notin 119868119889

N sub

119889

⋃

119894=1

k 119896119894 notin [minus119873119894 119873119894] (50)

6 Abstract and Applied Analysis

A straightforward computation shows that for 119905119894 isin [0 1V119894]

( sum

119896119894notin(minus119873119894 119873119894]

|sinc (V119894119905119894 minus 119896119894)|119902)

1119902

le (119862 sum

119896119894notin(minus119873119894 119873119894]

1

|119896|119902)

1119902

le (119862int

infin

119873119894

119905minus119902)

1119902

le 119862119873minus1119901

119894

(51)

Therefore

1003816100381610038161003816119878k (119879119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816 le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894)

(52)

It follows from (46) and (52) that

1003816100381610038161003816(119879119904

u119891 minus 119878k119891) (t)1003816100381610038161003816

le 119862((

119889

prod

119894=1

(2119873119894 + 1))

11199010

120590minus119892(r)+1119901

119901119889

0

+1205901119901

(

119889

sum

119894=1

119873minus1119901

119894))

(53)

We choose119873119894 = [(120590119892(r)

)119901

+ 1] and 1199010 = sum119889

119894=1ln(2119873119894 + 1) It is

easy to see that 1199010 gt 1 and (prod119889

119894=1(2119873119894 + 1))

11199010= 119890 A simple

computation gives

119901119889

0le 119862(ln

119889

prod

119894=1

119873119894)

119889

le 119862 ln119889120590 (54)

Note that119873119894 ge 120590119892(r)119901 Thus we have

119889

sum

119894=1

119873minus1119901

119894le 119862120590

minus119892(r) (55)

Collecting the above results we obtain

1003816100381610038161003816(119879119904

u119891) (t) minus (119878k119891) (t)1003816100381610038161003816 le 119862 sdot 120590

minus119892(r)+1119901ln119889120590 (56)

Combining (44) and (56) we prove the theorem

Proof of Theorem 2 By the triangle inequality we have

(119864kN119891) (t) le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816

+1003816100381610038161003816(119878k119891) (t) minus (119878kN119891) (t)

1003816100381610038161003816

(57)

By the arguments similar to those used in the proof ofTheorem 1 we obtain

1003816100381610038161003816(119878k119891) (t) minus (119878kN119891) (t)1003816100381610038161003816

le ( sum

ksdottminusknotin119868119889N

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

sdot ( sum

ksdottminusknotin119868119889N

100381610038161003816100381610038161003816sinc (k sdot t minus k)

100381610038161003816100381610038161003816

119902

)

1119902

le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894)

le 119862120590minus119892(r)+1119901ln119889120590

(58)

where we use119873119894 = [(120590119892(r)

)119901

+ 1] in the last inequalityCombiningTheorem 1 and (58) we complete the proof of

Theorem 2

3 The Measured Sampled Values Case

In practice the sampled values of a signal may not be exactlythe functional values and may have to be quantized Typicalerrors arising from these facts are jitter errors and amplitudeerrors Using the key idea of quasi-interpolation whichadopts integer translations of a basic function and integertranslations of a linear functional to approximate functionssee [8 25] and the references therein We may considersampled values that are the results of a linear functional andits integer translations acting on an undergoing signal [4 25]Such sampled values are called measured sampled valuesbecause they are closer to the truemeasurements taken fromasignalThe sampling series with themeasured sampled valuesis defined to be

(119878120582

k119891) (t) = sum

kisinZ119889120582119896119891(sdot +

kk) sinc (k sdot t minus k) (59)

where 120582 = 120582119896119896isinZ is any sequence of continuous linearfunctionals 1198620(R

119889) rarr C with 1198620(R

119889) being the set of all

continuous functions defined on R119889 and tending to zero atinfinity

Similar to the definition of (119878kN119891)(t) and (119864kN119891)(t) wehave the finite sum

(119878120582

kN119891) (t) = sum

kminusksdottisin119868119889N

120582119896119891(sdot +kk) sinc (k sdot t minus k) (60)

and the truncation error

(119864120582

kN119891) (t) =10038161003816100381610038161003816119891 (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816 (61)

To establish our theorems we need the error modulus

Ωk (119891 120582) = sup119896isinZ

10038161003816100381610038161003816100381610038161003816

120582119896119891(sdot +kk) minus 119891(

kk)

10038161003816100381610038161003816100381610038161003816

k gt 0 (62)

Abstract and Applied Analysis 7

WewriteΩ(119891 120582) forΩk(119891 120582) if no confusion arisesThe errormodulusΩ(119891 120582) provides a quantity for the quality of signalrsquosmeasured sampling values When the functionals in 120582 areconcrete we may get some reasonable estimates forΩ(119891 120582)

Sampling series with measured sampled values has beenstudied in [8] for bandlimited functions but without trunca-tionThe truncation errors are considered for functions fromLipschitz class with a decay condition in [13] Now we recalla typical result in [13]

Denote by Lip119871(1 119862(R119889

)) the set of all continuous func-tions 119891 satisfying

1003816100381610038161003816119891 (x) minus 119891 (y)1003816100381610038161003816 le 119871 (10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119909119889 minus 119910119889

1003816100381610038161003816) (63)

for all x = (1199091 119909119889) and y = (1199101 119910119889) isin R119889 SetkV = (1198961V 119896119889V) V sdot t = (V1199051 V119905119889) and tinfin =

max1199051 119905119889

Theorem D Let 119891 isin Lip119871(1 119862(R119889

)) satisfy the decay condi-tion

1003816100381610038161003816119891 (t)1003816100381610038161003816 le 119872119891tminus120574

infin tinfin ge 1 (64)

for some 0 lt 120574 le 1 Let 120582 = 120582119896 be any sequence of continuouslinear functionals For each V ge 119890

2 one has for the truncationerror at t isin R119889

10038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (t) minus sum

kisinZ119889kle119873120582119896119891(

kV) sinc (V sdot t minus k)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862Vminus1ln119889V

(65)

provided119873 = lfloor(V1+1120574minus1)2rfloor where lfloor119909rfloor is the smallest integerthat is greater or equal to a given 119909 isin R and Ω(119891 120582) le 1198880V

minus1

In [9] the author obtain the uniform bound of symmetrictruncation error for functions from isotropic Besov spacewith a similar decay condition Now we will provide theestimation for the truncation error

(119864120582

kN119891) (t) =10038161003816100381610038161003816119891 (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816

(66)

without any assumption about the decay of 119891 isin 119861r119901120579(R119889

)

Theorem 10 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define k as in Theorem B Let120582 = 120582119896 be any sequence of continuous linear functionals Fork one has

(119864120582

kN119891) (t) le 119862 sdot 120590minus119892(r)+1119901ln119889120590 (67)

provided 119873119894 = [(120590119892(r)

)119901

+ 1] and Ω(119891 120582) le 1198620120590minus119892(r)+1119901 for

some constant 1198620 gt 0

Proof By the triangle inequality we have

(119864120582

kN119891) (t)

le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 +10038161003816100381610038161003816(119878k119891) (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 + 1198681 + 1198682

(68)

where

1198681 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminusknotin119868119889N

119891(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198682 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminuskisin119868119889N

(119891(kk) minus 120582119896119891(sdot +

kk)) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)

Similar to (52) we have

1198681 le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894) (70)

Using Holderrsquos inequality we obtain

1198682 le ( sum

ksdottminuskisin119868119889N

10038161003816100381610038161003816100381610038161003816

119891 (kk) minus 120582119896119891(sdot +

kk)

10038161003816100381610038161003816100381610038161003816

1199010

)

11199010

sdot ( sum

ksdottminuskisin119868119889N

|sinc (k sdot t minus k) |1199020)11199020

le 119862(

119889

prod

119894=1

(2119873119894 + 1))

11199010

Ω(119891 120582) sdot 119901119889

0

(71)

where 11199010+11199020 = 1 Nowwe select the same119873119894 and1199010 as inthe proof ofTheorem 1 Similar to (55) we havesum119889

119894=1119873

minus1119901

119894le

119862120590minus119892(r) Thus

1198681 le 119862120590minus119892(r)+1119901

(72)

A simple computation gives (prod119889

119894=1(2119873119894 + 1))

11199010= 119890 and 1199010 le

119862 lnprod119889

119894=1119873119894 le 119862 ln120590 Notice that Ω(119891 120582) le 1198620120590

minus119892(r)+1119901Collecting these results we obtain

1198682 le 119862120590minus119892(r)+1119901ln119889120590 (73)

It follows fromTheorem 1 (72) and (73) that

(119864120582

kN119891) (t) le 119862120590minus119892(r)+1119901ln119889120590 (74)

which completes the proof

Finally we apply Theorem 10 to some practical examplesThe first one is that the measured sampled values are given byaverages of a function For119891 isin 1198620(R

119889)wedefine themodulus

of continuity

120596 (119891 120591) = suphle120591

1003817100381710038171003817119891 (sdot + h) minus 119891 (sdot)1003817100381710038171003817infin

(75)

where 120591may be any positive number

8 Abstract and Applied Analysis

Corollary 11 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define kN as in Theorem 10Suppose the sampled values 119891k of 119891 are obtained by the rule

119891k =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119891(t + kk)119889t (76)

where ℎ119895 are numbers satisfying 0 lt ℎ119895 le 120591 for all k isin Z119889

and 120591 gt 0 If 120591 lt 12 and 120596(119891 120591) le min119890119889(minus119892(r)+1119901)1198620120590

minus119892(r)+1119901 then

(119864120582

kN119891) (t) le 119862 sdot 120596(119891 120591)119892(r)minus1119901ln119889 1

120596 (119891 120591) (77)

Proof Let 120582 = 120582119896119896isinZ be the sequence of the linearfunctionals on 1198620(R

119889) and 119892 a continuous function on 119868

119889

h Define

120582119896119892 =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119892(t + kk)119889t (78)

Then 119891k = 120582119896119891(sdot + kk) Clearly |119891(t + kv) minus 119891(kv)| le

120596(119891 120591) Therefore

Ωk (119891 120582)

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816

120582119896119891(t + kk) minus 119891(

kk)

10038161003816100381610038161003816100381610038161003816

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

(119891(t + kk) minus 119891(

kk))119889t

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

120596 (119891 120591) 119889t10038161003816100381610038161003816100381610038161003816100381610038161003816

= 120596 (119891 120591)

(79)

Note that the function 119909 997891rarr 119909119892(r)minus1119901ln119889(1119909) is monotonely

increasing for 119909 isin (0 119890119889(minus119892(r)+1119901)

) The corollary followsfromTheorem 10

The second example is an estimate for the combination ofall four errors existing in sampling series the amplitude errorthe time-jitter error the truncation errors and the aliasingerrors We give some explanation for the amplitude error andthe time-jitter error

We assume the amplitude error results from quantizationwhich means the functional value 119891(119905) of a function 119891 atmoment 119905 is replaced by the nearest discrete value ormachinenumber 119891(119905) The quantization size is often known beforehand or can be chosen arbitrarily We may assume that thelocal error at any moment 119905 is bounded by a constant 120576 gt 0that is |119891(119905) minus 119891(119905)| le 120576 The time-jitter error arises if thesampled instances are notmet correctly butmight differ fromthe exact ones by 120591

1015840

119896 119896 isin Z we assume |1205911015840

119896| le 120591

1015840 for all 119896 andsome constant 1205911015840 gt 0 The combined error is defined to be

(119864119873119891) (119905) = 119891 (119905) minus sum

Vsdot119905minus119896isin[minus119873119873]

119891(119896

V+ 120591

1015840

119896) sinc (V119905 minus 119896)

(80)

Corollary 12 Let 119891 isin U(119861119903

119901120579(R)) 1 lt 119901 lt infin 1 le 120579 le infin

119903 gt 1 and V gt 119890 Then1003817100381710038171003817119864119873119891

1003817100381710038171003817infinle 119862Vminus119903+1119901 ln V (81)

provided119873 = [V119903119901+1] |119891(119905)minus119891(119905)| le 1198881Vminus119903+1119901 and120596(119891 120591

1015840) le

1198882Vminus119903+1119901 where 1198881 1198882 are positive constants and 1198881 + 1198882 le 1198620

Proof We define

120582119896 =

119891 (119896V + 1205911015840

119896)

119891 (119896V + 1205911015840

119896)120575 (sdot minus 120591

1015840

119896) (82)

where 120575 is the Dirac distribution Then 120582 = 120582119896119896isinZ is asequence of linear functional on1198620(R) It is clear that 120582119896119891(sdot+119896V) = 119891(119896V + 120591

1015840

119896) Then

10038161003816100381610038161003816100381610038161003816

120582119896119891(sdot +119896

V) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V+ 120591

1015840

119896)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le (1198881 + 1198882) Vminus119903+1119901

(83)

Thus Ω(119891 120582) le 1198620Vminus119903+1119901 By Theorem 10 we get the desired

result

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 10971251 11101220 and11271199) and the Program forNewCentury Excellent Talentsat University of China (NCET-10-0513)

References

[1] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 1948

[2] P L ButzerW Engels and U Scheben ldquoMagnitude of the trun-cation error in sam-pling expansions of bandlimited signalsrdquoIEEE Transactions on Acoustics Speech and Signal Processingvol 30 no 6 pp 906ndash912 1982

[3] A I ZayedAdvances in Shannonrsquos SamplingTheory CRC PressBoca Raton Fla USA 1993

[4] SDCasey andD FWalnut ldquoSystems of convolution equationsdeconvolution Shannon sampling and the wavelet and Gabortransformsrdquo SIAM Review vol 36 no 4 pp 537ndash577 1994

[5] P L Butzer G Schmeisser and R L Stens ldquoAn introductionto sampling analysisrdquo in Nonuniform Sampling Theory andPractice F Marvasti Ed pp 17ndash121 Kluwer Academic NewYork NY USA 2001

[6] S Smale and D-X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquo Bulletin of the AmericanMathematical Society vol 41 no 3 pp 279ndash305 2004

[7] S Smale and D-X Zhou ldquoShannon sampling II Connectionsto learning theoryrdquo Applied and Computational HarmonicAnalysis vol 19 no 3 pp 285ndash302 2005

Abstract and Applied Analysis 9

[8] P L Butzer and J Lei ldquoApproximation of signals usingmeasuredsampled values and error analysisrdquo Communications in AppliedAnalysis vol 4 no 2 pp 245ndash255 2000

[9] P X Ye ldquoError analysis for Shannon sampling series approxima-tionwithmeasured sampled valuesrdquoResearch Journal of AppliedSciences Engineering and Technology vol 5 no 3 pp 858ndash8642013

[10] J J Wang and G S Fang ldquoA multidimensional sampling theo-rem and an estimate of the aliasing errorrdquo Acta MathematicaeApplicatae Sinica vol 19 no 4 pp 481ndash488 1996

[11] X M Li ldquoUniform bounds for sampling expansionsrdquo Journal ofApproximation Theory vol 93 no 1 pp 100ndash113 1998

[12] L Jingfan and F Gensun ldquoOn uniform truncation error boundsand aliasing error for multidimensional sampling expansionrdquoSampling Theory in Signal and Image Processing vol 2 no 2pp 103ndash115 2003

[13] P L Butzer and J Lei ldquoErrors in truncated sampling serieswith measured sampled values for not-necessarily bandlimitedfunctionsrdquo Functiones et Approximatio vol 26 pp 25ndash39 1998

[14] H D Helms and J B Thomas ldquoTruncation error of sampling-theorem expansionsrdquo Proceedings of The IRE vol 50 no 2 pp179ndash184 1962

[15] D Jagerman ldquoBounds for truncation error of the samplingexpansionrdquo SIAM Journal on Applied Mathematics vol 14 no4 pp 714ndash723 1966

[16] C A Micchelli Y Xu and H Zhang ldquoOptimal learning ofbandlimited functions from localized samplingrdquo Journal ofComplexity vol 25 no 2 pp 85ndash114 2009

[17] A Ya Olenko and T K Pogany ldquoUniversal truncation errorupper bounds in sampling restorationrdquo Georgian MathematicalJournal vol 17 no 4 pp 765ndash786 2010

[18] P-X Ye and Z-H Song ldquoTruncation and aliasing errors forWhittaker-Kotelnikov-Shannon sampling expansionrdquo AppliedMathematics B vol 27 no 4 pp 412ndash418 2012

[19] P X Ye B H Sheng and X H Yuan ldquoOptimal order oftruncation and aliasing errors formulti-dimensional whittaker-shannon sampling expansionrdquo International Journal of Wirelessand Mobile Computing vol 5 no 4 pp 327ndash333 2012

[20] S M Nikolskii Approximation of Functions of Several Variablesand Imbedding Theorems Springer New York NY USA 1975

[21] Y Jiang and Y Liu ldquoAverage widths and optimal recovery ofmultivariate Besov classes in 119871119901(119877

119889)rdquo Journal of Approximation

Theory vol 102 no 1 pp 155ndash170 2000[22] C A Micchelli Y S Xu and P X Ye ldquoCucker-Smale learning

theory in Besov spacesrdquo in Advances in LearnIng TheoryMethods Models and Applications J Suykens G HorvathS Basu et al Eds pp 47ndash68 IOS Press Amsterdam TheNetherlands 2003

[23] R P Boas Jr Entire Functions Academic Press New York NYUSA 1954

[24] G Fang F J Hickernell and H Li ldquoApproximation onanisotropic Besov classes with mixed norms by standard infor-mationrdquo Journal of Complexity vol 21 no 3 pp 294ndash313 2005

[25] H G Burchard and J Lei ldquoCoordinate order of approxima-tion by functional-based approximation operatorsrdquo Journal ofApproximation Theory vol 82 no 2 pp 240ndash256 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Lemma 8 (see [10]) Let 119891 isin 119882119897

119901(R119889

) 1 le 119901 lt infin and 119897 ge 119889Then

(

119889

prod

119894=1

1

V119894sum

kisinZ119889

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

le 119862(1003817100381710038171003817119891

1003817100381710038171003817119871119901+

119889

sum

119894=1

1

V119894

10038171003817100381710038171003817100381710038171003817

120597119891

120597119909119894

10038171003817100381710038171003817100381710038171003817119871119901

+ sum

1le119894le119895le119889

1

V119894

1

V119895

1003817100381710038171003817100381710038171003817100381710038171003817

1205972119891

120597119909119894120597119909119895

1003817100381710038171003817100381710038171003817100381710038171003817119871119901

+ sdot sdot sdot +

119889

prod

119894=1

1

V119894

100381710038171003817100381710038171003817100381710038171003817

120597119889119891

1205971199091 sdot sdot sdot 120597119909119889

100381710038171003817100381710038171003817100381710038171003817119871119901

)

(38)

Lemma 9 (see [20 24]) Let l = (1198971 119897119889) isin N119889 r =

(1199031 119903119889) isin R119889

+ 1 le 119901 120579 le infin 1 minus sum

119889

119894=1119897119894119903119894 gt 0 and

r1015840 = (1 minus sum119889

119894=1119897119894119903119894)r For 119891 isin 119861

r119901120579(R119889

) it follows that thereexists a constant119862 depending on r r1015840 119901 and 120579 but independentof 119891 such that

10038171003817100381710038171003817120597|l|11989110038171003817100381710038171003817119861r1015840119901120579

le 1198621003817100381710038171003817119891

1003817100381710038171003817119861r119901120579

(39)

Proof of Theorem 1 It is known from Lemma 9 (letting 119897119894 = 1

for 119894 isin 119885119889) that the fact 119891 isin U(119861rp120579(R

119889)) with 119903119894 gt 119889 119894 isin 119885119889

implies 119891 isin 119882119889

119901(R119889

) therefore by Lemma 810038171003817100381710038171003817100381710038171003817

119891(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

1119901 (40)

And hence by Lemma 5

(119878k119891) (t) = sum

kisinZ119889119891(

kk) sinc (k sdot t minus k) (41)

converges uniformly on R119889Set u = k2119904 and 2119904 gt 119889 +max119903119894 119894 = 1 119889 So 119879

119904

u119891 isin

119861119901

k (R119889) as mentioned above By Theorem A we have 119879

119904

u119891 =

119878k(119879119904

u119891) Thus

(119879119904

u119891) (t) minus (119878k119891) (t)

= sum

kisinZ119889((119879

119904

u119891)(kk) minus 119891(

kk)) sinc (k sdot t minus k)

(42)

Using the triangle inequality we obtain1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 +

1003816100381610038161003816(119879119904

u119891) (t) minus (119878k119891) (t)1003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 +

1003816100381610038161003816119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

+1003816100381610038161003816119878k (119879

119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

(43)

By Lemma 31003816100381610038161003816119891 (t) minus (119879

119904

u119891) (t)1003816100381610038161003816 le 119862120590

minus119892(r)+1119901 (44)

It is clear that1003816100381610038161003816119878kN (119879

119904

u119891 minus 119891) (t)1003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminuskisin119868119889N

(119879119904

u119891 minus 119891)(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(45)

Applying Holderrsquos inequality with exponent 1199010 we get1003816100381610038161003816119878kN (119879

119904

u119891 minus 119891) (t)1003816100381610038161003816

le ( sum

ksdottminuskisin119868119889N

10038161003816100381610038161003816100381610038161003816

(119879119904

u119891 minus 119891)(kk)

10038161003816100381610038161003816100381610038161003816

1199010

)

11199010

sdot ( sum

ksdottminuskisin119868119889N

|sinc(k sdot t minus k)|1199020)11199020

le 119862(

119889

prod

119894=1

2119873119894 + 1)

11199010

120590minus119892(r)+1119901

sdot 119901119889

0

(46)

where 11199010+11199020 = 1 and the second inequality follows from(44) and Lemma 6

Next we estimate |119878k(119879119904

u119891 minus 119891)(t) minus 119878kN(119879119904

u119891 minus 119891)(t)| ByHolderrsquos inequality

1003816100381610038161003816119878k (119879119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816

le ( sum

ksdottminusknotin119868119889N

10038161003816100381610038161003816100381610038161003816

(119879119904

u119891 minus 119891)(kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

sdot ( sum

ksdottminusknotin119868119889N

100381610038161003816100381610038161003816sinc(k sdot t minus k)

100381610038161003816100381610038161003816

119902

)

1119902

(47)

where 1119901 + 1119902 = 1By Lemma 7 and Lemma 4 we obtain

10038171003817100381710038171003817100381710038171003817

(119879119904

u119891)(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

11199011003817100381710038171003817119879119904

u1198911003817100381710038171003817119871119901

le 11986212059011199011003817100381710038171003817119891

1003817100381710038171003817119871119901le 119862120590

1119901

(48)

It follows from (40) (48) and Minkowski inequality10038171003817100381710038171003817100381710038171003817

(119879119904

u119891 minus 119891)(kk)

10038171003817100381710038171003817100381710038171003817119897119901le 119862120590

1119901 (49)

Set ℎ(t) = (sumksdottminusknotin119868119889N|sinc(k sdot t minus k)|119902)1119902 Note that ℎ(t +

mk) = ℎ(t) for all t isin R119889 and m isin Z119889 Thus to give anupper estimate for ℎ(t) on R119889 we only need to bound it onprod

119889

119894=1[0 1V119894] Note that

k k notin 119868119889

N sub

119889

⋃

119894=1

k 119896119894 notin [minus119873119894 119873119894] (50)

6 Abstract and Applied Analysis

A straightforward computation shows that for 119905119894 isin [0 1V119894]

( sum

119896119894notin(minus119873119894 119873119894]

|sinc (V119894119905119894 minus 119896119894)|119902)

1119902

le (119862 sum

119896119894notin(minus119873119894 119873119894]

1

|119896|119902)

1119902

le (119862int

infin

119873119894

119905minus119902)

1119902

le 119862119873minus1119901

119894

(51)

Therefore

1003816100381610038161003816119878k (119879119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816 le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894)

(52)

It follows from (46) and (52) that

1003816100381610038161003816(119879119904

u119891 minus 119878k119891) (t)1003816100381610038161003816

le 119862((

119889

prod

119894=1

(2119873119894 + 1))

11199010

120590minus119892(r)+1119901

119901119889

0

+1205901119901

(

119889

sum

119894=1

119873minus1119901

119894))

(53)

We choose119873119894 = [(120590119892(r)

)119901

+ 1] and 1199010 = sum119889

119894=1ln(2119873119894 + 1) It is

easy to see that 1199010 gt 1 and (prod119889

119894=1(2119873119894 + 1))

11199010= 119890 A simple

computation gives

119901119889

0le 119862(ln

119889

prod

119894=1

119873119894)

119889

le 119862 ln119889120590 (54)

Note that119873119894 ge 120590119892(r)119901 Thus we have

119889

sum

119894=1

119873minus1119901

119894le 119862120590

minus119892(r) (55)

Collecting the above results we obtain

1003816100381610038161003816(119879119904

u119891) (t) minus (119878k119891) (t)1003816100381610038161003816 le 119862 sdot 120590

minus119892(r)+1119901ln119889120590 (56)

Combining (44) and (56) we prove the theorem

Proof of Theorem 2 By the triangle inequality we have

(119864kN119891) (t) le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816

+1003816100381610038161003816(119878k119891) (t) minus (119878kN119891) (t)

1003816100381610038161003816

(57)

By the arguments similar to those used in the proof ofTheorem 1 we obtain

1003816100381610038161003816(119878k119891) (t) minus (119878kN119891) (t)1003816100381610038161003816

le ( sum

ksdottminusknotin119868119889N

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

sdot ( sum

ksdottminusknotin119868119889N

100381610038161003816100381610038161003816sinc (k sdot t minus k)

100381610038161003816100381610038161003816

119902

)

1119902

le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894)

le 119862120590minus119892(r)+1119901ln119889120590

(58)

where we use119873119894 = [(120590119892(r)

)119901

+ 1] in the last inequalityCombiningTheorem 1 and (58) we complete the proof of

Theorem 2

3 The Measured Sampled Values Case

In practice the sampled values of a signal may not be exactlythe functional values and may have to be quantized Typicalerrors arising from these facts are jitter errors and amplitudeerrors Using the key idea of quasi-interpolation whichadopts integer translations of a basic function and integertranslations of a linear functional to approximate functionssee [8 25] and the references therein We may considersampled values that are the results of a linear functional andits integer translations acting on an undergoing signal [4 25]Such sampled values are called measured sampled valuesbecause they are closer to the truemeasurements taken fromasignalThe sampling series with themeasured sampled valuesis defined to be

(119878120582

k119891) (t) = sum

kisinZ119889120582119896119891(sdot +

kk) sinc (k sdot t minus k) (59)

where 120582 = 120582119896119896isinZ is any sequence of continuous linearfunctionals 1198620(R

119889) rarr C with 1198620(R

119889) being the set of all

continuous functions defined on R119889 and tending to zero atinfinity

Similar to the definition of (119878kN119891)(t) and (119864kN119891)(t) wehave the finite sum

(119878120582

kN119891) (t) = sum

kminusksdottisin119868119889N

120582119896119891(sdot +kk) sinc (k sdot t minus k) (60)

and the truncation error

(119864120582

kN119891) (t) =10038161003816100381610038161003816119891 (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816 (61)

To establish our theorems we need the error modulus

Ωk (119891 120582) = sup119896isinZ

10038161003816100381610038161003816100381610038161003816

120582119896119891(sdot +kk) minus 119891(

kk)

10038161003816100381610038161003816100381610038161003816

k gt 0 (62)

Abstract and Applied Analysis 7

WewriteΩ(119891 120582) forΩk(119891 120582) if no confusion arisesThe errormodulusΩ(119891 120582) provides a quantity for the quality of signalrsquosmeasured sampling values When the functionals in 120582 areconcrete we may get some reasonable estimates forΩ(119891 120582)

Sampling series with measured sampled values has beenstudied in [8] for bandlimited functions but without trunca-tionThe truncation errors are considered for functions fromLipschitz class with a decay condition in [13] Now we recalla typical result in [13]

Denote by Lip119871(1 119862(R119889

)) the set of all continuous func-tions 119891 satisfying

1003816100381610038161003816119891 (x) minus 119891 (y)1003816100381610038161003816 le 119871 (10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119909119889 minus 119910119889

1003816100381610038161003816) (63)

for all x = (1199091 119909119889) and y = (1199101 119910119889) isin R119889 SetkV = (1198961V 119896119889V) V sdot t = (V1199051 V119905119889) and tinfin =

max1199051 119905119889

Theorem D Let 119891 isin Lip119871(1 119862(R119889

)) satisfy the decay condi-tion

1003816100381610038161003816119891 (t)1003816100381610038161003816 le 119872119891tminus120574

infin tinfin ge 1 (64)

for some 0 lt 120574 le 1 Let 120582 = 120582119896 be any sequence of continuouslinear functionals For each V ge 119890

2 one has for the truncationerror at t isin R119889

10038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (t) minus sum

kisinZ119889kle119873120582119896119891(

kV) sinc (V sdot t minus k)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862Vminus1ln119889V

(65)

provided119873 = lfloor(V1+1120574minus1)2rfloor where lfloor119909rfloor is the smallest integerthat is greater or equal to a given 119909 isin R and Ω(119891 120582) le 1198880V

minus1

In [9] the author obtain the uniform bound of symmetrictruncation error for functions from isotropic Besov spacewith a similar decay condition Now we will provide theestimation for the truncation error

(119864120582

kN119891) (t) =10038161003816100381610038161003816119891 (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816

(66)

without any assumption about the decay of 119891 isin 119861r119901120579(R119889

)

Theorem 10 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define k as in Theorem B Let120582 = 120582119896 be any sequence of continuous linear functionals Fork one has

(119864120582

kN119891) (t) le 119862 sdot 120590minus119892(r)+1119901ln119889120590 (67)

provided 119873119894 = [(120590119892(r)

)119901

+ 1] and Ω(119891 120582) le 1198620120590minus119892(r)+1119901 for

some constant 1198620 gt 0

Proof By the triangle inequality we have

(119864120582

kN119891) (t)

le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 +10038161003816100381610038161003816(119878k119891) (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 + 1198681 + 1198682

(68)

where

1198681 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminusknotin119868119889N

119891(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198682 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminuskisin119868119889N

(119891(kk) minus 120582119896119891(sdot +

kk)) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)

Similar to (52) we have

1198681 le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894) (70)

Using Holderrsquos inequality we obtain

1198682 le ( sum

ksdottminuskisin119868119889N

10038161003816100381610038161003816100381610038161003816

119891 (kk) minus 120582119896119891(sdot +

kk)

10038161003816100381610038161003816100381610038161003816

1199010

)

11199010

sdot ( sum

ksdottminuskisin119868119889N

|sinc (k sdot t minus k) |1199020)11199020

le 119862(

119889

prod

119894=1

(2119873119894 + 1))

11199010

Ω(119891 120582) sdot 119901119889

0

(71)

where 11199010+11199020 = 1 Nowwe select the same119873119894 and1199010 as inthe proof ofTheorem 1 Similar to (55) we havesum119889

119894=1119873

minus1119901

119894le

119862120590minus119892(r) Thus

1198681 le 119862120590minus119892(r)+1119901

(72)

A simple computation gives (prod119889

119894=1(2119873119894 + 1))

11199010= 119890 and 1199010 le

119862 lnprod119889

119894=1119873119894 le 119862 ln120590 Notice that Ω(119891 120582) le 1198620120590

minus119892(r)+1119901Collecting these results we obtain

1198682 le 119862120590minus119892(r)+1119901ln119889120590 (73)

It follows fromTheorem 1 (72) and (73) that

(119864120582

kN119891) (t) le 119862120590minus119892(r)+1119901ln119889120590 (74)

which completes the proof

Finally we apply Theorem 10 to some practical examplesThe first one is that the measured sampled values are given byaverages of a function For119891 isin 1198620(R

119889)wedefine themodulus

of continuity

120596 (119891 120591) = suphle120591

1003817100381710038171003817119891 (sdot + h) minus 119891 (sdot)1003817100381710038171003817infin

(75)

where 120591may be any positive number

8 Abstract and Applied Analysis

Corollary 11 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define kN as in Theorem 10Suppose the sampled values 119891k of 119891 are obtained by the rule

119891k =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119891(t + kk)119889t (76)

where ℎ119895 are numbers satisfying 0 lt ℎ119895 le 120591 for all k isin Z119889

and 120591 gt 0 If 120591 lt 12 and 120596(119891 120591) le min119890119889(minus119892(r)+1119901)1198620120590

minus119892(r)+1119901 then

(119864120582

kN119891) (t) le 119862 sdot 120596(119891 120591)119892(r)minus1119901ln119889 1

120596 (119891 120591) (77)

Proof Let 120582 = 120582119896119896isinZ be the sequence of the linearfunctionals on 1198620(R

119889) and 119892 a continuous function on 119868

119889

h Define

120582119896119892 =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119892(t + kk)119889t (78)

Then 119891k = 120582119896119891(sdot + kk) Clearly |119891(t + kv) minus 119891(kv)| le

120596(119891 120591) Therefore

Ωk (119891 120582)

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816

120582119896119891(t + kk) minus 119891(

kk)

10038161003816100381610038161003816100381610038161003816

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

(119891(t + kk) minus 119891(

kk))119889t

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

120596 (119891 120591) 119889t10038161003816100381610038161003816100381610038161003816100381610038161003816

= 120596 (119891 120591)

(79)

Note that the function 119909 997891rarr 119909119892(r)minus1119901ln119889(1119909) is monotonely

increasing for 119909 isin (0 119890119889(minus119892(r)+1119901)

) The corollary followsfromTheorem 10

The second example is an estimate for the combination ofall four errors existing in sampling series the amplitude errorthe time-jitter error the truncation errors and the aliasingerrors We give some explanation for the amplitude error andthe time-jitter error

We assume the amplitude error results from quantizationwhich means the functional value 119891(119905) of a function 119891 atmoment 119905 is replaced by the nearest discrete value ormachinenumber 119891(119905) The quantization size is often known beforehand or can be chosen arbitrarily We may assume that thelocal error at any moment 119905 is bounded by a constant 120576 gt 0that is |119891(119905) minus 119891(119905)| le 120576 The time-jitter error arises if thesampled instances are notmet correctly butmight differ fromthe exact ones by 120591

1015840

119896 119896 isin Z we assume |1205911015840

119896| le 120591

1015840 for all 119896 andsome constant 1205911015840 gt 0 The combined error is defined to be

(119864119873119891) (119905) = 119891 (119905) minus sum

Vsdot119905minus119896isin[minus119873119873]

119891(119896

V+ 120591

1015840

119896) sinc (V119905 minus 119896)

(80)

Corollary 12 Let 119891 isin U(119861119903

119901120579(R)) 1 lt 119901 lt infin 1 le 120579 le infin

119903 gt 1 and V gt 119890 Then1003817100381710038171003817119864119873119891

1003817100381710038171003817infinle 119862Vminus119903+1119901 ln V (81)

provided119873 = [V119903119901+1] |119891(119905)minus119891(119905)| le 1198881Vminus119903+1119901 and120596(119891 120591

1015840) le

1198882Vminus119903+1119901 where 1198881 1198882 are positive constants and 1198881 + 1198882 le 1198620

Proof We define

120582119896 =

119891 (119896V + 1205911015840

119896)

119891 (119896V + 1205911015840

119896)120575 (sdot minus 120591

1015840

119896) (82)

where 120575 is the Dirac distribution Then 120582 = 120582119896119896isinZ is asequence of linear functional on1198620(R) It is clear that 120582119896119891(sdot+119896V) = 119891(119896V + 120591

1015840

119896) Then

10038161003816100381610038161003816100381610038161003816

120582119896119891(sdot +119896

V) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V+ 120591

1015840

119896)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le (1198881 + 1198882) Vminus119903+1119901

(83)

Thus Ω(119891 120582) le 1198620Vminus119903+1119901 By Theorem 10 we get the desired

result

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 10971251 11101220 and11271199) and the Program forNewCentury Excellent Talentsat University of China (NCET-10-0513)

References

[1] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 1948

[2] P L ButzerW Engels and U Scheben ldquoMagnitude of the trun-cation error in sam-pling expansions of bandlimited signalsrdquoIEEE Transactions on Acoustics Speech and Signal Processingvol 30 no 6 pp 906ndash912 1982

[3] A I ZayedAdvances in Shannonrsquos SamplingTheory CRC PressBoca Raton Fla USA 1993

[4] SDCasey andD FWalnut ldquoSystems of convolution equationsdeconvolution Shannon sampling and the wavelet and Gabortransformsrdquo SIAM Review vol 36 no 4 pp 537ndash577 1994

[5] P L Butzer G Schmeisser and R L Stens ldquoAn introductionto sampling analysisrdquo in Nonuniform Sampling Theory andPractice F Marvasti Ed pp 17ndash121 Kluwer Academic NewYork NY USA 2001

[6] S Smale and D-X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquo Bulletin of the AmericanMathematical Society vol 41 no 3 pp 279ndash305 2004

[7] S Smale and D-X Zhou ldquoShannon sampling II Connectionsto learning theoryrdquo Applied and Computational HarmonicAnalysis vol 19 no 3 pp 285ndash302 2005

Abstract and Applied Analysis 9

[8] P L Butzer and J Lei ldquoApproximation of signals usingmeasuredsampled values and error analysisrdquo Communications in AppliedAnalysis vol 4 no 2 pp 245ndash255 2000

[9] P X Ye ldquoError analysis for Shannon sampling series approxima-tionwithmeasured sampled valuesrdquoResearch Journal of AppliedSciences Engineering and Technology vol 5 no 3 pp 858ndash8642013

[10] J J Wang and G S Fang ldquoA multidimensional sampling theo-rem and an estimate of the aliasing errorrdquo Acta MathematicaeApplicatae Sinica vol 19 no 4 pp 481ndash488 1996

[11] X M Li ldquoUniform bounds for sampling expansionsrdquo Journal ofApproximation Theory vol 93 no 1 pp 100ndash113 1998

[12] L Jingfan and F Gensun ldquoOn uniform truncation error boundsand aliasing error for multidimensional sampling expansionrdquoSampling Theory in Signal and Image Processing vol 2 no 2pp 103ndash115 2003

[13] P L Butzer and J Lei ldquoErrors in truncated sampling serieswith measured sampled values for not-necessarily bandlimitedfunctionsrdquo Functiones et Approximatio vol 26 pp 25ndash39 1998

[14] H D Helms and J B Thomas ldquoTruncation error of sampling-theorem expansionsrdquo Proceedings of The IRE vol 50 no 2 pp179ndash184 1962

[15] D Jagerman ldquoBounds for truncation error of the samplingexpansionrdquo SIAM Journal on Applied Mathematics vol 14 no4 pp 714ndash723 1966

[16] C A Micchelli Y Xu and H Zhang ldquoOptimal learning ofbandlimited functions from localized samplingrdquo Journal ofComplexity vol 25 no 2 pp 85ndash114 2009

[17] A Ya Olenko and T K Pogany ldquoUniversal truncation errorupper bounds in sampling restorationrdquo Georgian MathematicalJournal vol 17 no 4 pp 765ndash786 2010

[18] P-X Ye and Z-H Song ldquoTruncation and aliasing errors forWhittaker-Kotelnikov-Shannon sampling expansionrdquo AppliedMathematics B vol 27 no 4 pp 412ndash418 2012

[19] P X Ye B H Sheng and X H Yuan ldquoOptimal order oftruncation and aliasing errors formulti-dimensional whittaker-shannon sampling expansionrdquo International Journal of Wirelessand Mobile Computing vol 5 no 4 pp 327ndash333 2012

[20] S M Nikolskii Approximation of Functions of Several Variablesand Imbedding Theorems Springer New York NY USA 1975

[21] Y Jiang and Y Liu ldquoAverage widths and optimal recovery ofmultivariate Besov classes in 119871119901(119877

119889)rdquo Journal of Approximation

Theory vol 102 no 1 pp 155ndash170 2000[22] C A Micchelli Y S Xu and P X Ye ldquoCucker-Smale learning

theory in Besov spacesrdquo in Advances in LearnIng TheoryMethods Models and Applications J Suykens G HorvathS Basu et al Eds pp 47ndash68 IOS Press Amsterdam TheNetherlands 2003

[23] R P Boas Jr Entire Functions Academic Press New York NYUSA 1954

[24] G Fang F J Hickernell and H Li ldquoApproximation onanisotropic Besov classes with mixed norms by standard infor-mationrdquo Journal of Complexity vol 21 no 3 pp 294ndash313 2005

[25] H G Burchard and J Lei ldquoCoordinate order of approxima-tion by functional-based approximation operatorsrdquo Journal ofApproximation Theory vol 82 no 2 pp 240ndash256 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

A straightforward computation shows that for 119905119894 isin [0 1V119894]

( sum

119896119894notin(minus119873119894 119873119894]

|sinc (V119894119905119894 minus 119896119894)|119902)

1119902

le (119862 sum

119896119894notin(minus119873119894 119873119894]

1

|119896|119902)

1119902

le (119862int

infin

119873119894

119905minus119902)

1119902

le 119862119873minus1119901

119894

(51)

Therefore

1003816100381610038161003816119878k (119879119904

u119891 minus 119891) (t) minus 119878kN (119879119904

u119891 minus 119891) (t)1003816100381610038161003816 le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894)

(52)

It follows from (46) and (52) that

1003816100381610038161003816(119879119904

u119891 minus 119878k119891) (t)1003816100381610038161003816

le 119862((

119889

prod

119894=1

(2119873119894 + 1))

11199010

120590minus119892(r)+1119901

119901119889

0

+1205901119901

(

119889

sum

119894=1

119873minus1119901

119894))

(53)

We choose119873119894 = [(120590119892(r)

)119901

+ 1] and 1199010 = sum119889

119894=1ln(2119873119894 + 1) It is

easy to see that 1199010 gt 1 and (prod119889

119894=1(2119873119894 + 1))

11199010= 119890 A simple

computation gives

119901119889

0le 119862(ln

119889

prod

119894=1

119873119894)

119889

le 119862 ln119889120590 (54)

Note that119873119894 ge 120590119892(r)119901 Thus we have

119889

sum

119894=1

119873minus1119901

119894le 119862120590

minus119892(r) (55)

Collecting the above results we obtain

1003816100381610038161003816(119879119904

u119891) (t) minus (119878k119891) (t)1003816100381610038161003816 le 119862 sdot 120590

minus119892(r)+1119901ln119889120590 (56)

Combining (44) and (56) we prove the theorem

Proof of Theorem 2 By the triangle inequality we have

(119864kN119891) (t) le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816

+1003816100381610038161003816(119878k119891) (t) minus (119878kN119891) (t)

1003816100381610038161003816

(57)

By the arguments similar to those used in the proof ofTheorem 1 we obtain

1003816100381610038161003816(119878k119891) (t) minus (119878kN119891) (t)1003816100381610038161003816

le ( sum

ksdottminusknotin119868119889N

10038161003816100381610038161003816100381610038161003816

119891 (kk)

10038161003816100381610038161003816100381610038161003816

119901

)

1119901

sdot ( sum

ksdottminusknotin119868119889N

100381610038161003816100381610038161003816sinc (k sdot t minus k)

100381610038161003816100381610038161003816

119902

)

1119902

le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894)

le 119862120590minus119892(r)+1119901ln119889120590

(58)

where we use119873119894 = [(120590119892(r)

)119901

+ 1] in the last inequalityCombiningTheorem 1 and (58) we complete the proof of

Theorem 2

3 The Measured Sampled Values Case

In practice the sampled values of a signal may not be exactlythe functional values and may have to be quantized Typicalerrors arising from these facts are jitter errors and amplitudeerrors Using the key idea of quasi-interpolation whichadopts integer translations of a basic function and integertranslations of a linear functional to approximate functionssee [8 25] and the references therein We may considersampled values that are the results of a linear functional andits integer translations acting on an undergoing signal [4 25]Such sampled values are called measured sampled valuesbecause they are closer to the truemeasurements taken fromasignalThe sampling series with themeasured sampled valuesis defined to be

(119878120582

k119891) (t) = sum

kisinZ119889120582119896119891(sdot +

kk) sinc (k sdot t minus k) (59)

where 120582 = 120582119896119896isinZ is any sequence of continuous linearfunctionals 1198620(R

119889) rarr C with 1198620(R

119889) being the set of all

continuous functions defined on R119889 and tending to zero atinfinity

Similar to the definition of (119878kN119891)(t) and (119864kN119891)(t) wehave the finite sum

(119878120582

kN119891) (t) = sum

kminusksdottisin119868119889N

120582119896119891(sdot +kk) sinc (k sdot t minus k) (60)

and the truncation error

(119864120582

kN119891) (t) =10038161003816100381610038161003816119891 (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816 (61)

To establish our theorems we need the error modulus

Ωk (119891 120582) = sup119896isinZ

10038161003816100381610038161003816100381610038161003816

120582119896119891(sdot +kk) minus 119891(

kk)

10038161003816100381610038161003816100381610038161003816

k gt 0 (62)

Abstract and Applied Analysis 7

WewriteΩ(119891 120582) forΩk(119891 120582) if no confusion arisesThe errormodulusΩ(119891 120582) provides a quantity for the quality of signalrsquosmeasured sampling values When the functionals in 120582 areconcrete we may get some reasonable estimates forΩ(119891 120582)

Sampling series with measured sampled values has beenstudied in [8] for bandlimited functions but without trunca-tionThe truncation errors are considered for functions fromLipschitz class with a decay condition in [13] Now we recalla typical result in [13]

Denote by Lip119871(1 119862(R119889

)) the set of all continuous func-tions 119891 satisfying

1003816100381610038161003816119891 (x) minus 119891 (y)1003816100381610038161003816 le 119871 (10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119909119889 minus 119910119889

1003816100381610038161003816) (63)

for all x = (1199091 119909119889) and y = (1199101 119910119889) isin R119889 SetkV = (1198961V 119896119889V) V sdot t = (V1199051 V119905119889) and tinfin =

max1199051 119905119889

Theorem D Let 119891 isin Lip119871(1 119862(R119889

)) satisfy the decay condi-tion

1003816100381610038161003816119891 (t)1003816100381610038161003816 le 119872119891tminus120574

infin tinfin ge 1 (64)

for some 0 lt 120574 le 1 Let 120582 = 120582119896 be any sequence of continuouslinear functionals For each V ge 119890

2 one has for the truncationerror at t isin R119889

10038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (t) minus sum

kisinZ119889kle119873120582119896119891(

kV) sinc (V sdot t minus k)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862Vminus1ln119889V

(65)

provided119873 = lfloor(V1+1120574minus1)2rfloor where lfloor119909rfloor is the smallest integerthat is greater or equal to a given 119909 isin R and Ω(119891 120582) le 1198880V

minus1

In [9] the author obtain the uniform bound of symmetrictruncation error for functions from isotropic Besov spacewith a similar decay condition Now we will provide theestimation for the truncation error

(119864120582

kN119891) (t) =10038161003816100381610038161003816119891 (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816

(66)

without any assumption about the decay of 119891 isin 119861r119901120579(R119889

)

Theorem 10 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define k as in Theorem B Let120582 = 120582119896 be any sequence of continuous linear functionals Fork one has

(119864120582

kN119891) (t) le 119862 sdot 120590minus119892(r)+1119901ln119889120590 (67)

provided 119873119894 = [(120590119892(r)

)119901

+ 1] and Ω(119891 120582) le 1198620120590minus119892(r)+1119901 for

some constant 1198620 gt 0

Proof By the triangle inequality we have

(119864120582

kN119891) (t)

le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 +10038161003816100381610038161003816(119878k119891) (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 + 1198681 + 1198682

(68)

where

1198681 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminusknotin119868119889N

119891(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198682 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminuskisin119868119889N

(119891(kk) minus 120582119896119891(sdot +

kk)) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)

Similar to (52) we have

1198681 le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894) (70)

Using Holderrsquos inequality we obtain

1198682 le ( sum

ksdottminuskisin119868119889N

10038161003816100381610038161003816100381610038161003816

119891 (kk) minus 120582119896119891(sdot +

kk)

10038161003816100381610038161003816100381610038161003816

1199010

)

11199010

sdot ( sum

ksdottminuskisin119868119889N

|sinc (k sdot t minus k) |1199020)11199020

le 119862(

119889

prod

119894=1

(2119873119894 + 1))

11199010

Ω(119891 120582) sdot 119901119889

0

(71)

where 11199010+11199020 = 1 Nowwe select the same119873119894 and1199010 as inthe proof ofTheorem 1 Similar to (55) we havesum119889

119894=1119873

minus1119901

119894le

119862120590minus119892(r) Thus

1198681 le 119862120590minus119892(r)+1119901

(72)

A simple computation gives (prod119889

119894=1(2119873119894 + 1))

11199010= 119890 and 1199010 le

119862 lnprod119889

119894=1119873119894 le 119862 ln120590 Notice that Ω(119891 120582) le 1198620120590

minus119892(r)+1119901Collecting these results we obtain

1198682 le 119862120590minus119892(r)+1119901ln119889120590 (73)

It follows fromTheorem 1 (72) and (73) that

(119864120582

kN119891) (t) le 119862120590minus119892(r)+1119901ln119889120590 (74)

which completes the proof

Finally we apply Theorem 10 to some practical examplesThe first one is that the measured sampled values are given byaverages of a function For119891 isin 1198620(R

119889)wedefine themodulus

of continuity

120596 (119891 120591) = suphle120591

1003817100381710038171003817119891 (sdot + h) minus 119891 (sdot)1003817100381710038171003817infin

(75)

where 120591may be any positive number

8 Abstract and Applied Analysis

Corollary 11 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define kN as in Theorem 10Suppose the sampled values 119891k of 119891 are obtained by the rule

119891k =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119891(t + kk)119889t (76)

where ℎ119895 are numbers satisfying 0 lt ℎ119895 le 120591 for all k isin Z119889

and 120591 gt 0 If 120591 lt 12 and 120596(119891 120591) le min119890119889(minus119892(r)+1119901)1198620120590

minus119892(r)+1119901 then

(119864120582

kN119891) (t) le 119862 sdot 120596(119891 120591)119892(r)minus1119901ln119889 1

120596 (119891 120591) (77)

Proof Let 120582 = 120582119896119896isinZ be the sequence of the linearfunctionals on 1198620(R

119889) and 119892 a continuous function on 119868

119889

h Define

120582119896119892 =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119892(t + kk)119889t (78)

Then 119891k = 120582119896119891(sdot + kk) Clearly |119891(t + kv) minus 119891(kv)| le

120596(119891 120591) Therefore

Ωk (119891 120582)

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816

120582119896119891(t + kk) minus 119891(

kk)

10038161003816100381610038161003816100381610038161003816

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

(119891(t + kk) minus 119891(

kk))119889t

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

120596 (119891 120591) 119889t10038161003816100381610038161003816100381610038161003816100381610038161003816

= 120596 (119891 120591)

(79)

Note that the function 119909 997891rarr 119909119892(r)minus1119901ln119889(1119909) is monotonely

increasing for 119909 isin (0 119890119889(minus119892(r)+1119901)

) The corollary followsfromTheorem 10

The second example is an estimate for the combination ofall four errors existing in sampling series the amplitude errorthe time-jitter error the truncation errors and the aliasingerrors We give some explanation for the amplitude error andthe time-jitter error

We assume the amplitude error results from quantizationwhich means the functional value 119891(119905) of a function 119891 atmoment 119905 is replaced by the nearest discrete value ormachinenumber 119891(119905) The quantization size is often known beforehand or can be chosen arbitrarily We may assume that thelocal error at any moment 119905 is bounded by a constant 120576 gt 0that is |119891(119905) minus 119891(119905)| le 120576 The time-jitter error arises if thesampled instances are notmet correctly butmight differ fromthe exact ones by 120591

1015840

119896 119896 isin Z we assume |1205911015840

119896| le 120591

1015840 for all 119896 andsome constant 1205911015840 gt 0 The combined error is defined to be

(119864119873119891) (119905) = 119891 (119905) minus sum

Vsdot119905minus119896isin[minus119873119873]

119891(119896

V+ 120591

1015840

119896) sinc (V119905 minus 119896)

(80)

Corollary 12 Let 119891 isin U(119861119903

119901120579(R)) 1 lt 119901 lt infin 1 le 120579 le infin

119903 gt 1 and V gt 119890 Then1003817100381710038171003817119864119873119891

1003817100381710038171003817infinle 119862Vminus119903+1119901 ln V (81)

provided119873 = [V119903119901+1] |119891(119905)minus119891(119905)| le 1198881Vminus119903+1119901 and120596(119891 120591

1015840) le

1198882Vminus119903+1119901 where 1198881 1198882 are positive constants and 1198881 + 1198882 le 1198620

Proof We define

120582119896 =

119891 (119896V + 1205911015840

119896)

119891 (119896V + 1205911015840

119896)120575 (sdot minus 120591

1015840

119896) (82)

where 120575 is the Dirac distribution Then 120582 = 120582119896119896isinZ is asequence of linear functional on1198620(R) It is clear that 120582119896119891(sdot+119896V) = 119891(119896V + 120591

1015840

119896) Then

10038161003816100381610038161003816100381610038161003816

120582119896119891(sdot +119896

V) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V+ 120591

1015840

119896)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le (1198881 + 1198882) Vminus119903+1119901

(83)

Thus Ω(119891 120582) le 1198620Vminus119903+1119901 By Theorem 10 we get the desired

result

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 10971251 11101220 and11271199) and the Program forNewCentury Excellent Talentsat University of China (NCET-10-0513)

References

[1] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 1948

[2] P L ButzerW Engels and U Scheben ldquoMagnitude of the trun-cation error in sam-pling expansions of bandlimited signalsrdquoIEEE Transactions on Acoustics Speech and Signal Processingvol 30 no 6 pp 906ndash912 1982

[3] A I ZayedAdvances in Shannonrsquos SamplingTheory CRC PressBoca Raton Fla USA 1993

[4] SDCasey andD FWalnut ldquoSystems of convolution equationsdeconvolution Shannon sampling and the wavelet and Gabortransformsrdquo SIAM Review vol 36 no 4 pp 537ndash577 1994

[5] P L Butzer G Schmeisser and R L Stens ldquoAn introductionto sampling analysisrdquo in Nonuniform Sampling Theory andPractice F Marvasti Ed pp 17ndash121 Kluwer Academic NewYork NY USA 2001

[6] S Smale and D-X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquo Bulletin of the AmericanMathematical Society vol 41 no 3 pp 279ndash305 2004

[7] S Smale and D-X Zhou ldquoShannon sampling II Connectionsto learning theoryrdquo Applied and Computational HarmonicAnalysis vol 19 no 3 pp 285ndash302 2005

Abstract and Applied Analysis 9

[8] P L Butzer and J Lei ldquoApproximation of signals usingmeasuredsampled values and error analysisrdquo Communications in AppliedAnalysis vol 4 no 2 pp 245ndash255 2000

[9] P X Ye ldquoError analysis for Shannon sampling series approxima-tionwithmeasured sampled valuesrdquoResearch Journal of AppliedSciences Engineering and Technology vol 5 no 3 pp 858ndash8642013

[10] J J Wang and G S Fang ldquoA multidimensional sampling theo-rem and an estimate of the aliasing errorrdquo Acta MathematicaeApplicatae Sinica vol 19 no 4 pp 481ndash488 1996

[11] X M Li ldquoUniform bounds for sampling expansionsrdquo Journal ofApproximation Theory vol 93 no 1 pp 100ndash113 1998

[12] L Jingfan and F Gensun ldquoOn uniform truncation error boundsand aliasing error for multidimensional sampling expansionrdquoSampling Theory in Signal and Image Processing vol 2 no 2pp 103ndash115 2003

[13] P L Butzer and J Lei ldquoErrors in truncated sampling serieswith measured sampled values for not-necessarily bandlimitedfunctionsrdquo Functiones et Approximatio vol 26 pp 25ndash39 1998

[14] H D Helms and J B Thomas ldquoTruncation error of sampling-theorem expansionsrdquo Proceedings of The IRE vol 50 no 2 pp179ndash184 1962

[15] D Jagerman ldquoBounds for truncation error of the samplingexpansionrdquo SIAM Journal on Applied Mathematics vol 14 no4 pp 714ndash723 1966

[16] C A Micchelli Y Xu and H Zhang ldquoOptimal learning ofbandlimited functions from localized samplingrdquo Journal ofComplexity vol 25 no 2 pp 85ndash114 2009

[17] A Ya Olenko and T K Pogany ldquoUniversal truncation errorupper bounds in sampling restorationrdquo Georgian MathematicalJournal vol 17 no 4 pp 765ndash786 2010

[18] P-X Ye and Z-H Song ldquoTruncation and aliasing errors forWhittaker-Kotelnikov-Shannon sampling expansionrdquo AppliedMathematics B vol 27 no 4 pp 412ndash418 2012

[19] P X Ye B H Sheng and X H Yuan ldquoOptimal order oftruncation and aliasing errors formulti-dimensional whittaker-shannon sampling expansionrdquo International Journal of Wirelessand Mobile Computing vol 5 no 4 pp 327ndash333 2012

[20] S M Nikolskii Approximation of Functions of Several Variablesand Imbedding Theorems Springer New York NY USA 1975

[21] Y Jiang and Y Liu ldquoAverage widths and optimal recovery ofmultivariate Besov classes in 119871119901(119877

119889)rdquo Journal of Approximation

Theory vol 102 no 1 pp 155ndash170 2000[22] C A Micchelli Y S Xu and P X Ye ldquoCucker-Smale learning

theory in Besov spacesrdquo in Advances in LearnIng TheoryMethods Models and Applications J Suykens G HorvathS Basu et al Eds pp 47ndash68 IOS Press Amsterdam TheNetherlands 2003

[23] R P Boas Jr Entire Functions Academic Press New York NYUSA 1954

[24] G Fang F J Hickernell and H Li ldquoApproximation onanisotropic Besov classes with mixed norms by standard infor-mationrdquo Journal of Complexity vol 21 no 3 pp 294ndash313 2005

[25] H G Burchard and J Lei ldquoCoordinate order of approxima-tion by functional-based approximation operatorsrdquo Journal ofApproximation Theory vol 82 no 2 pp 240ndash256 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

WewriteΩ(119891 120582) forΩk(119891 120582) if no confusion arisesThe errormodulusΩ(119891 120582) provides a quantity for the quality of signalrsquosmeasured sampling values When the functionals in 120582 areconcrete we may get some reasonable estimates forΩ(119891 120582)

Sampling series with measured sampled values has beenstudied in [8] for bandlimited functions but without trunca-tionThe truncation errors are considered for functions fromLipschitz class with a decay condition in [13] Now we recalla typical result in [13]

Denote by Lip119871(1 119862(R119889

)) the set of all continuous func-tions 119891 satisfying

1003816100381610038161003816119891 (x) minus 119891 (y)1003816100381610038161003816 le 119871 (10038161003816100381610038161199091 minus 1199101

1003816100381610038161003816 + sdot sdot sdot +1003816100381610038161003816119909119889 minus 119910119889

1003816100381610038161003816) (63)

for all x = (1199091 119909119889) and y = (1199101 119910119889) isin R119889 SetkV = (1198961V 119896119889V) V sdot t = (V1199051 V119905119889) and tinfin =

max1199051 119905119889

Theorem D Let 119891 isin Lip119871(1 119862(R119889

)) satisfy the decay condi-tion

1003816100381610038161003816119891 (t)1003816100381610038161003816 le 119872119891tminus120574

infin tinfin ge 1 (64)

for some 0 lt 120574 le 1 Let 120582 = 120582119896 be any sequence of continuouslinear functionals For each V ge 119890

2 one has for the truncationerror at t isin R119889

10038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (t) minus sum

kisinZ119889kle119873120582119896119891(

kV) sinc (V sdot t minus k)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862Vminus1ln119889V

(65)

provided119873 = lfloor(V1+1120574minus1)2rfloor where lfloor119909rfloor is the smallest integerthat is greater or equal to a given 119909 isin R and Ω(119891 120582) le 1198880V

minus1

In [9] the author obtain the uniform bound of symmetrictruncation error for functions from isotropic Besov spacewith a similar decay condition Now we will provide theestimation for the truncation error

(119864120582

kN119891) (t) =10038161003816100381610038161003816119891 (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816

(66)

without any assumption about the decay of 119891 isin 119861r119901120579(R119889

)

Theorem 10 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define k as in Theorem B Let120582 = 120582119896 be any sequence of continuous linear functionals Fork one has

(119864120582

kN119891) (t) le 119862 sdot 120590minus119892(r)+1119901ln119889120590 (67)

provided 119873119894 = [(120590119892(r)

)119901

+ 1] and Ω(119891 120582) le 1198620120590minus119892(r)+1119901 for

some constant 1198620 gt 0

Proof By the triangle inequality we have

(119864120582

kN119891) (t)

le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 +10038161003816100381610038161003816(119878k119891) (t) minus (119878

120582

kN119891) (t)10038161003816100381610038161003816

le1003816100381610038161003816119891 (t) minus (119878k119891) (t)

1003816100381610038161003816 + 1198681 + 1198682

(68)

where

1198681 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminusknotin119868119889N

119891(kk) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198682 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sum

ksdottminuskisin119868119889N

(119891(kk) minus 120582119896119891(sdot +

kk)) sinc (k sdot t minus k)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(69)

Similar to (52) we have

1198681 le 1198621205901119901

(

119889

sum

119894=1

119873minus1119901

119894) (70)

Using Holderrsquos inequality we obtain

1198682 le ( sum

ksdottminuskisin119868119889N

10038161003816100381610038161003816100381610038161003816

119891 (kk) minus 120582119896119891(sdot +

kk)

10038161003816100381610038161003816100381610038161003816

1199010

)

11199010

sdot ( sum

ksdottminuskisin119868119889N

|sinc (k sdot t minus k) |1199020)11199020

le 119862(

119889

prod

119894=1

(2119873119894 + 1))

11199010

Ω(119891 120582) sdot 119901119889

0

(71)

where 11199010+11199020 = 1 Nowwe select the same119873119894 and1199010 as inthe proof ofTheorem 1 Similar to (55) we havesum119889

119894=1119873

minus1119901

119894le

119862120590minus119892(r) Thus

1198681 le 119862120590minus119892(r)+1119901

(72)

A simple computation gives (prod119889

119894=1(2119873119894 + 1))

11199010= 119890 and 1199010 le

119862 lnprod119889

119894=1119873119894 le 119862 ln120590 Notice that Ω(119891 120582) le 1198620120590

minus119892(r)+1119901Collecting these results we obtain

1198682 le 119862120590minus119892(r)+1119901ln119889120590 (73)

It follows fromTheorem 1 (72) and (73) that

(119864120582

kN119891) (t) le 119862120590minus119892(r)+1119901ln119889120590 (74)

which completes the proof

Finally we apply Theorem 10 to some practical examplesThe first one is that the measured sampled values are given byaverages of a function For119891 isin 1198620(R

119889)wedefine themodulus

of continuity

120596 (119891 120591) = suphle120591

1003817100381710038171003817119891 (sdot + h) minus 119891 (sdot)1003817100381710038171003817infin

(75)

where 120591may be any positive number

8 Abstract and Applied Analysis

Corollary 11 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define kN as in Theorem 10Suppose the sampled values 119891k of 119891 are obtained by the rule

119891k =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119891(t + kk)119889t (76)

where ℎ119895 are numbers satisfying 0 lt ℎ119895 le 120591 for all k isin Z119889

and 120591 gt 0 If 120591 lt 12 and 120596(119891 120591) le min119890119889(minus119892(r)+1119901)1198620120590

minus119892(r)+1119901 then

(119864120582

kN119891) (t) le 119862 sdot 120596(119891 120591)119892(r)minus1119901ln119889 1

120596 (119891 120591) (77)

Proof Let 120582 = 120582119896119896isinZ be the sequence of the linearfunctionals on 1198620(R

119889) and 119892 a continuous function on 119868

119889

h Define

120582119896119892 =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119892(t + kk)119889t (78)

Then 119891k = 120582119896119891(sdot + kk) Clearly |119891(t + kv) minus 119891(kv)| le

120596(119891 120591) Therefore

Ωk (119891 120582)

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816

120582119896119891(t + kk) minus 119891(

kk)

10038161003816100381610038161003816100381610038161003816

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

(119891(t + kk) minus 119891(

kk))119889t

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

120596 (119891 120591) 119889t10038161003816100381610038161003816100381610038161003816100381610038161003816

= 120596 (119891 120591)

(79)

Note that the function 119909 997891rarr 119909119892(r)minus1119901ln119889(1119909) is monotonely

increasing for 119909 isin (0 119890119889(minus119892(r)+1119901)

) The corollary followsfromTheorem 10

The second example is an estimate for the combination ofall four errors existing in sampling series the amplitude errorthe time-jitter error the truncation errors and the aliasingerrors We give some explanation for the amplitude error andthe time-jitter error

We assume the amplitude error results from quantizationwhich means the functional value 119891(119905) of a function 119891 atmoment 119905 is replaced by the nearest discrete value ormachinenumber 119891(119905) The quantization size is often known beforehand or can be chosen arbitrarily We may assume that thelocal error at any moment 119905 is bounded by a constant 120576 gt 0that is |119891(119905) minus 119891(119905)| le 120576 The time-jitter error arises if thesampled instances are notmet correctly butmight differ fromthe exact ones by 120591

1015840

119896 119896 isin Z we assume |1205911015840

119896| le 120591

1015840 for all 119896 andsome constant 1205911015840 gt 0 The combined error is defined to be

(119864119873119891) (119905) = 119891 (119905) minus sum

Vsdot119905minus119896isin[minus119873119873]

119891(119896

V+ 120591

1015840

119896) sinc (V119905 minus 119896)

(80)

Corollary 12 Let 119891 isin U(119861119903

119901120579(R)) 1 lt 119901 lt infin 1 le 120579 le infin

119903 gt 1 and V gt 119890 Then1003817100381710038171003817119864119873119891

1003817100381710038171003817infinle 119862Vminus119903+1119901 ln V (81)

provided119873 = [V119903119901+1] |119891(119905)minus119891(119905)| le 1198881Vminus119903+1119901 and120596(119891 120591

1015840) le

1198882Vminus119903+1119901 where 1198881 1198882 are positive constants and 1198881 + 1198882 le 1198620

Proof We define

120582119896 =

119891 (119896V + 1205911015840

119896)

119891 (119896V + 1205911015840

119896)120575 (sdot minus 120591

1015840

119896) (82)

where 120575 is the Dirac distribution Then 120582 = 120582119896119896isinZ is asequence of linear functional on1198620(R) It is clear that 120582119896119891(sdot+119896V) = 119891(119896V + 120591

1015840

119896) Then

10038161003816100381610038161003816100381610038161003816

120582119896119891(sdot +119896

V) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V+ 120591

1015840

119896)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le (1198881 + 1198882) Vminus119903+1119901

(83)

Thus Ω(119891 120582) le 1198620Vminus119903+1119901 By Theorem 10 we get the desired

result

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 10971251 11101220 and11271199) and the Program forNewCentury Excellent Talentsat University of China (NCET-10-0513)

References

[1] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 1948

[2] P L ButzerW Engels and U Scheben ldquoMagnitude of the trun-cation error in sam-pling expansions of bandlimited signalsrdquoIEEE Transactions on Acoustics Speech and Signal Processingvol 30 no 6 pp 906ndash912 1982

[3] A I ZayedAdvances in Shannonrsquos SamplingTheory CRC PressBoca Raton Fla USA 1993

[4] SDCasey andD FWalnut ldquoSystems of convolution equationsdeconvolution Shannon sampling and the wavelet and Gabortransformsrdquo SIAM Review vol 36 no 4 pp 537ndash577 1994

[5] P L Butzer G Schmeisser and R L Stens ldquoAn introductionto sampling analysisrdquo in Nonuniform Sampling Theory andPractice F Marvasti Ed pp 17ndash121 Kluwer Academic NewYork NY USA 2001

[6] S Smale and D-X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquo Bulletin of the AmericanMathematical Society vol 41 no 3 pp 279ndash305 2004

[7] S Smale and D-X Zhou ldquoShannon sampling II Connectionsto learning theoryrdquo Applied and Computational HarmonicAnalysis vol 19 no 3 pp 285ndash302 2005

Abstract and Applied Analysis 9

[8] P L Butzer and J Lei ldquoApproximation of signals usingmeasuredsampled values and error analysisrdquo Communications in AppliedAnalysis vol 4 no 2 pp 245ndash255 2000

[9] P X Ye ldquoError analysis for Shannon sampling series approxima-tionwithmeasured sampled valuesrdquoResearch Journal of AppliedSciences Engineering and Technology vol 5 no 3 pp 858ndash8642013

[10] J J Wang and G S Fang ldquoA multidimensional sampling theo-rem and an estimate of the aliasing errorrdquo Acta MathematicaeApplicatae Sinica vol 19 no 4 pp 481ndash488 1996

[11] X M Li ldquoUniform bounds for sampling expansionsrdquo Journal ofApproximation Theory vol 93 no 1 pp 100ndash113 1998

[12] L Jingfan and F Gensun ldquoOn uniform truncation error boundsand aliasing error for multidimensional sampling expansionrdquoSampling Theory in Signal and Image Processing vol 2 no 2pp 103ndash115 2003

[13] P L Butzer and J Lei ldquoErrors in truncated sampling serieswith measured sampled values for not-necessarily bandlimitedfunctionsrdquo Functiones et Approximatio vol 26 pp 25ndash39 1998

[14] H D Helms and J B Thomas ldquoTruncation error of sampling-theorem expansionsrdquo Proceedings of The IRE vol 50 no 2 pp179ndash184 1962

[15] D Jagerman ldquoBounds for truncation error of the samplingexpansionrdquo SIAM Journal on Applied Mathematics vol 14 no4 pp 714ndash723 1966

[16] C A Micchelli Y Xu and H Zhang ldquoOptimal learning ofbandlimited functions from localized samplingrdquo Journal ofComplexity vol 25 no 2 pp 85ndash114 2009

[17] A Ya Olenko and T K Pogany ldquoUniversal truncation errorupper bounds in sampling restorationrdquo Georgian MathematicalJournal vol 17 no 4 pp 765ndash786 2010

[18] P-X Ye and Z-H Song ldquoTruncation and aliasing errors forWhittaker-Kotelnikov-Shannon sampling expansionrdquo AppliedMathematics B vol 27 no 4 pp 412ndash418 2012

[19] P X Ye B H Sheng and X H Yuan ldquoOptimal order oftruncation and aliasing errors formulti-dimensional whittaker-shannon sampling expansionrdquo International Journal of Wirelessand Mobile Computing vol 5 no 4 pp 327ndash333 2012

[20] S M Nikolskii Approximation of Functions of Several Variablesand Imbedding Theorems Springer New York NY USA 1975

[21] Y Jiang and Y Liu ldquoAverage widths and optimal recovery ofmultivariate Besov classes in 119871119901(119877

119889)rdquo Journal of Approximation

Theory vol 102 no 1 pp 155ndash170 2000[22] C A Micchelli Y S Xu and P X Ye ldquoCucker-Smale learning

theory in Besov spacesrdquo in Advances in LearnIng TheoryMethods Models and Applications J Suykens G HorvathS Basu et al Eds pp 47ndash68 IOS Press Amsterdam TheNetherlands 2003

[23] R P Boas Jr Entire Functions Academic Press New York NYUSA 1954

[24] G Fang F J Hickernell and H Li ldquoApproximation onanisotropic Besov classes with mixed norms by standard infor-mationrdquo Journal of Complexity vol 21 no 3 pp 294ndash313 2005

[25] H G Burchard and J Lei ldquoCoordinate order of approxima-tion by functional-based approximation operatorsrdquo Journal ofApproximation Theory vol 82 no 2 pp 240ndash256 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

Corollary 11 Let 119891 isin U(119861r119901120579(R119889

)) with the same 119901 120579 andr as in Theorem 1 For 120590 gt 119890 define kN as in Theorem 10Suppose the sampled values 119891k of 119891 are obtained by the rule

119891k =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119891(t + kk)119889t (76)

where ℎ119895 are numbers satisfying 0 lt ℎ119895 le 120591 for all k isin Z119889

and 120591 gt 0 If 120591 lt 12 and 120596(119891 120591) le min119890119889(minus119892(r)+1119901)1198620120590

minus119892(r)+1119901 then

(119864120582

kN119891) (t) le 119862 sdot 120596(119891 120591)119892(r)minus1119901ln119889 1

120596 (119891 120591) (77)

Proof Let 120582 = 120582119896119896isinZ be the sequence of the linearfunctionals on 1198620(R

119889) and 119892 a continuous function on 119868

119889

h Define

120582119896119892 =1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

119892(t + kk)119889t (78)

Then 119891k = 120582119896119891(sdot + kk) Clearly |119891(t + kv) minus 119891(kv)| le

120596(119891 120591) Therefore

Ωk (119891 120582)

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816

120582119896119891(t + kk) minus 119891(

kk)

10038161003816100381610038161003816100381610038161003816

= sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

(119891(t + kk) minus 119891(

kk))119889t

10038161003816100381610038161003816100381610038161003816100381610038161003816

le sup119896isinZ

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

2119889

119889

prod

119895=1

1

ℎ119895

int119868119889h

120596 (119891 120591) 119889t10038161003816100381610038161003816100381610038161003816100381610038161003816

= 120596 (119891 120591)

(79)

Note that the function 119909 997891rarr 119909119892(r)minus1119901ln119889(1119909) is monotonely

increasing for 119909 isin (0 119890119889(minus119892(r)+1119901)

) The corollary followsfromTheorem 10

The second example is an estimate for the combination ofall four errors existing in sampling series the amplitude errorthe time-jitter error the truncation errors and the aliasingerrors We give some explanation for the amplitude error andthe time-jitter error

We assume the amplitude error results from quantizationwhich means the functional value 119891(119905) of a function 119891 atmoment 119905 is replaced by the nearest discrete value ormachinenumber 119891(119905) The quantization size is often known beforehand or can be chosen arbitrarily We may assume that thelocal error at any moment 119905 is bounded by a constant 120576 gt 0that is |119891(119905) minus 119891(119905)| le 120576 The time-jitter error arises if thesampled instances are notmet correctly butmight differ fromthe exact ones by 120591

1015840

119896 119896 isin Z we assume |1205911015840

119896| le 120591

1015840 for all 119896 andsome constant 1205911015840 gt 0 The combined error is defined to be

(119864119873119891) (119905) = 119891 (119905) minus sum

Vsdot119905minus119896isin[minus119873119873]

119891(119896

V+ 120591

1015840

119896) sinc (V119905 minus 119896)

(80)

Corollary 12 Let 119891 isin U(119861119903

119901120579(R)) 1 lt 119901 lt infin 1 le 120579 le infin

119903 gt 1 and V gt 119890 Then1003817100381710038171003817119864119873119891

1003817100381710038171003817infinle 119862Vminus119903+1119901 ln V (81)

provided119873 = [V119903119901+1] |119891(119905)minus119891(119905)| le 1198881Vminus119903+1119901 and120596(119891 120591

1015840) le

1198882Vminus119903+1119901 where 1198881 1198882 are positive constants and 1198881 + 1198882 le 1198620

Proof We define

120582119896 =

119891 (119896V + 1205911015840

119896)

119891 (119896V + 1205911015840

119896)120575 (sdot minus 120591

1015840

119896) (82)

where 120575 is the Dirac distribution Then 120582 = 120582119896119896isinZ is asequence of linear functional on1198620(R) It is clear that 120582119896119891(sdot+119896V) = 119891(119896V + 120591

1015840

119896) Then

10038161003816100381610038161003816100381610038161003816

120582119896119891(sdot +119896

V) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V+ 120591

1015840

119896)

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

119891 (119896

V+ 120591

1015840

119896) minus 119891(

119896

V)

10038161003816100381610038161003816100381610038161003816

le (1198881 + 1198882) Vminus119903+1119901

(83)

Thus Ω(119891 120582) le 1198620Vminus119903+1119901 By Theorem 10 we get the desired

result

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 10971251 11101220 and11271199) and the Program forNewCentury Excellent Talentsat University of China (NCET-10-0513)

References

[1] C E Shannon ldquoAmathematical theory of communicationrdquoTheBell System Technical Journal vol 27 pp 379ndash423 1948

[2] P L ButzerW Engels and U Scheben ldquoMagnitude of the trun-cation error in sam-pling expansions of bandlimited signalsrdquoIEEE Transactions on Acoustics Speech and Signal Processingvol 30 no 6 pp 906ndash912 1982

[3] A I ZayedAdvances in Shannonrsquos SamplingTheory CRC PressBoca Raton Fla USA 1993

[4] SDCasey andD FWalnut ldquoSystems of convolution equationsdeconvolution Shannon sampling and the wavelet and Gabortransformsrdquo SIAM Review vol 36 no 4 pp 537ndash577 1994

[5] P L Butzer G Schmeisser and R L Stens ldquoAn introductionto sampling analysisrdquo in Nonuniform Sampling Theory andPractice F Marvasti Ed pp 17ndash121 Kluwer Academic NewYork NY USA 2001

[6] S Smale and D-X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquo Bulletin of the AmericanMathematical Society vol 41 no 3 pp 279ndash305 2004

[7] S Smale and D-X Zhou ldquoShannon sampling II Connectionsto learning theoryrdquo Applied and Computational HarmonicAnalysis vol 19 no 3 pp 285ndash302 2005

Abstract and Applied Analysis 9

[8] P L Butzer and J Lei ldquoApproximation of signals usingmeasuredsampled values and error analysisrdquo Communications in AppliedAnalysis vol 4 no 2 pp 245ndash255 2000

[9] P X Ye ldquoError analysis for Shannon sampling series approxima-tionwithmeasured sampled valuesrdquoResearch Journal of AppliedSciences Engineering and Technology vol 5 no 3 pp 858ndash8642013

[10] J J Wang and G S Fang ldquoA multidimensional sampling theo-rem and an estimate of the aliasing errorrdquo Acta MathematicaeApplicatae Sinica vol 19 no 4 pp 481ndash488 1996

[11] X M Li ldquoUniform bounds for sampling expansionsrdquo Journal ofApproximation Theory vol 93 no 1 pp 100ndash113 1998

[12] L Jingfan and F Gensun ldquoOn uniform truncation error boundsand aliasing error for multidimensional sampling expansionrdquoSampling Theory in Signal and Image Processing vol 2 no 2pp 103ndash115 2003

[13] P L Butzer and J Lei ldquoErrors in truncated sampling serieswith measured sampled values for not-necessarily bandlimitedfunctionsrdquo Functiones et Approximatio vol 26 pp 25ndash39 1998

[14] H D Helms and J B Thomas ldquoTruncation error of sampling-theorem expansionsrdquo Proceedings of The IRE vol 50 no 2 pp179ndash184 1962

[15] D Jagerman ldquoBounds for truncation error of the samplingexpansionrdquo SIAM Journal on Applied Mathematics vol 14 no4 pp 714ndash723 1966

[16] C A Micchelli Y Xu and H Zhang ldquoOptimal learning ofbandlimited functions from localized samplingrdquo Journal ofComplexity vol 25 no 2 pp 85ndash114 2009

[17] A Ya Olenko and T K Pogany ldquoUniversal truncation errorupper bounds in sampling restorationrdquo Georgian MathematicalJournal vol 17 no 4 pp 765ndash786 2010

[18] P-X Ye and Z-H Song ldquoTruncation and aliasing errors forWhittaker-Kotelnikov-Shannon sampling expansionrdquo AppliedMathematics B vol 27 no 4 pp 412ndash418 2012

[19] P X Ye B H Sheng and X H Yuan ldquoOptimal order oftruncation and aliasing errors formulti-dimensional whittaker-shannon sampling expansionrdquo International Journal of Wirelessand Mobile Computing vol 5 no 4 pp 327ndash333 2012

[20] S M Nikolskii Approximation of Functions of Several Variablesand Imbedding Theorems Springer New York NY USA 1975

[21] Y Jiang and Y Liu ldquoAverage widths and optimal recovery ofmultivariate Besov classes in 119871119901(119877

119889)rdquo Journal of Approximation

Theory vol 102 no 1 pp 155ndash170 2000[22] C A Micchelli Y S Xu and P X Ye ldquoCucker-Smale learning

theory in Besov spacesrdquo in Advances in LearnIng TheoryMethods Models and Applications J Suykens G HorvathS Basu et al Eds pp 47ndash68 IOS Press Amsterdam TheNetherlands 2003

[23] R P Boas Jr Entire Functions Academic Press New York NYUSA 1954

[24] G Fang F J Hickernell and H Li ldquoApproximation onanisotropic Besov classes with mixed norms by standard infor-mationrdquo Journal of Complexity vol 21 no 3 pp 294ndash313 2005

[25] H G Burchard and J Lei ldquoCoordinate order of approxima-tion by functional-based approximation operatorsrdquo Journal ofApproximation Theory vol 82 no 2 pp 240ndash256 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

[8] P L Butzer and J Lei ldquoApproximation of signals usingmeasuredsampled values and error analysisrdquo Communications in AppliedAnalysis vol 4 no 2 pp 245ndash255 2000

[9] P X Ye ldquoError analysis for Shannon sampling series approxima-tionwithmeasured sampled valuesrdquoResearch Journal of AppliedSciences Engineering and Technology vol 5 no 3 pp 858ndash8642013

[10] J J Wang and G S Fang ldquoA multidimensional sampling theo-rem and an estimate of the aliasing errorrdquo Acta MathematicaeApplicatae Sinica vol 19 no 4 pp 481ndash488 1996

[11] X M Li ldquoUniform bounds for sampling expansionsrdquo Journal ofApproximation Theory vol 93 no 1 pp 100ndash113 1998

[12] L Jingfan and F Gensun ldquoOn uniform truncation error boundsand aliasing error for multidimensional sampling expansionrdquoSampling Theory in Signal and Image Processing vol 2 no 2pp 103ndash115 2003

[13] P L Butzer and J Lei ldquoErrors in truncated sampling serieswith measured sampled values for not-necessarily bandlimitedfunctionsrdquo Functiones et Approximatio vol 26 pp 25ndash39 1998

[14] H D Helms and J B Thomas ldquoTruncation error of sampling-theorem expansionsrdquo Proceedings of The IRE vol 50 no 2 pp179ndash184 1962

[15] D Jagerman ldquoBounds for truncation error of the samplingexpansionrdquo SIAM Journal on Applied Mathematics vol 14 no4 pp 714ndash723 1966

[16] C A Micchelli Y Xu and H Zhang ldquoOptimal learning ofbandlimited functions from localized samplingrdquo Journal ofComplexity vol 25 no 2 pp 85ndash114 2009

[17] A Ya Olenko and T K Pogany ldquoUniversal truncation errorupper bounds in sampling restorationrdquo Georgian MathematicalJournal vol 17 no 4 pp 765ndash786 2010

[18] P-X Ye and Z-H Song ldquoTruncation and aliasing errors forWhittaker-Kotelnikov-Shannon sampling expansionrdquo AppliedMathematics B vol 27 no 4 pp 412ndash418 2012

[19] P X Ye B H Sheng and X H Yuan ldquoOptimal order oftruncation and aliasing errors formulti-dimensional whittaker-shannon sampling expansionrdquo International Journal of Wirelessand Mobile Computing vol 5 no 4 pp 327ndash333 2012

[20] S M Nikolskii Approximation of Functions of Several Variablesand Imbedding Theorems Springer New York NY USA 1975

[21] Y Jiang and Y Liu ldquoAverage widths and optimal recovery ofmultivariate Besov classes in 119871119901(119877

119889)rdquo Journal of Approximation

Theory vol 102 no 1 pp 155ndash170 2000[22] C A Micchelli Y S Xu and P X Ye ldquoCucker-Smale learning

theory in Besov spacesrdquo in Advances in LearnIng TheoryMethods Models and Applications J Suykens G HorvathS Basu et al Eds pp 47ndash68 IOS Press Amsterdam TheNetherlands 2003

[23] R P Boas Jr Entire Functions Academic Press New York NYUSA 1954

[24] G Fang F J Hickernell and H Li ldquoApproximation onanisotropic Besov classes with mixed norms by standard infor-mationrdquo Journal of Complexity vol 21 no 3 pp 294ndash313 2005

[25] H G Burchard and J Lei ldquoCoordinate order of approxima-tion by functional-based approximation operatorsrdquo Journal ofApproximation Theory vol 82 no 2 pp 240ndash256 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of