approximation of unbounded functions with linear positive operators
TRANSCRIPT
APPROXIMATION OF UNBOUNDED FUNCTIONS WITH LINEAR POSITIVE OPERATORS
R. K. S. RATHORE
DELFTSE UNIVERSITAIRE PERS
Approximation of unbounded functions with linear positive operators
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Approximation of unbounded functions with linear positive operators
PROEFSCHRIFT ter verkrijging van de graad van doctor in de technische wetenschappen aan de Technische Hogeschool Delft, op gezag van de rector magnificus ir. H. B. Boerema, hoogleraar in de afdeling der elektrotechniek, voor een commissie aangewezen door het college van dekanen te verdedigen op woensdag 27 november 1974 te 14.00 uur door
RAM KISHORE SINGH RATHORE
Ph. D. in Mathematics (Indian Institute of Technology, Delhi) geboren te Lakhimpur-Kheri (U.P.) India
1974/Delftse Universitaire Pers
r
Dit proefschrift is goedgekeurd door de promotor PROF. DR. P. C. SIKKEMA
k
CONTENTS
INTRODUCTION ^ 1
CHAPTER I METHOD OF TEST FUNCTIONS 8
1.1 Approximation of functions having at the most
a polynomial growth when the variable tends to
+ 00 8
1.1.1 Definitions and notations 8
1.1.2 Convergence of the operators L 10
1.1.3 Asymptotic formulae for twice differentiable
functions I8
1.1.i* A class of linear positive operator sequences 32
1.1.5 Generalizations for functions of several vari
ables Ul
1.1.6 On a generalized sequence of linear positive
operators 55
1.1.T A method of constructing operators for func
tions of several variables 61
1.2 The trigonometric case 63
1.2.1 Asymptotic formulae for twice differentiable
functions 6h
1.2.2 Generalizations for functions of several va
riables 79
1.3 Approximation of functions of an exponential
growth 86
1.3.1 Single variable case 87
1.3.2 Generalizations for functions of several va
riables 91
V
CHAPTER 2 METHOD OF BOUNDING FUNCTIONS AND THE W-FUNCTIONS 97
2.1 Method of bounding functions 97
2.1.1 Basic convergence 97
2.1.2 Asymptotic estimates 99
2.1.3 Asymptotic formulae 1OU
2.2 Combining the techniques of bounding functions
and the W-functions 111
2.2.1 Asymptotic formulae 111
2.2.2 Asymptotic estimates 117
CHAPTER 3 APPROXIMATION OF UNBOUNDED FUNCTIONS BY OPERA
TORS OF SUMMATION TYPE 121
3.1 A general outline 121
3.2 The Bernstein polynomials 122
CHAPTER k APPROXIMATION OF UNBOUNDED FUNCTION<=! BY OPERA
TORS OF INTEGRAL TYPE 133
h.^ Approximation of \mbounded integrable functions 139
i+.l.l The general method 139
U.I.2 The generalized Jackson operators L lUo np-p
U.I.3 The Gamma operators G lU2
U.1.U Singular integrals W of Gauss-Weierstrass lU6
U.I.5 De La Vallee - Poussin integrals V lU8
REFERENCES 151
SAI.IENVATTING 16o
VI
INTRODUCTION
1. Schurer [ 59-62] , Hsu [ 17-18] , Wang [ 18] , Wood [ 12-14]
llttller [48] Eisenberg [l2-14]and several other
researchers have studied the possibility of
approximating a real (or complex) valued fiinction f(t),
defined on the real line or on a subset of it and
unbounded as t — + oo (or to some other point or points),
at its points of continuity by means of a suitable
sequence JL } (n=1,2,...) of linear operators ultimately
positive for the points (or a single point) of continuity.
In general, such a procedure assumes the convergence
L (g;x) — g(x) as n — oo, where x is a fixed point ot it
belongs to the set of points on which an approximation
is desired, for some test fvmctions g(t) and, in
addition, the uniform boundedness (uniform with respect
to n) or a kind of convergence as n -» oo ) of the
sequence JL (Q;X)J (n=1,2,...), where Q(t) is a suitably
chosen unbounded function. Under such assumptions we
arrive at a class (for which Q(t) is called a bounding
function) of unbounded functions f(t) which can be
aporoximated at a point t=x of continuity by the
senuence JL (f;x)j as n ->• oo . In this connection we
remark that almost all the work of above authors does
not use the full force of their assumptions and that
it is possible to enlar en the classes of unbounded
fljinctions for \jixich their results hold under the same or
even milder assumptions.
Recently l/alk [81] and Schmid [56] used the notion
of a socalled \/-function h(t), defined for t £ [0,oo )
and satisfying h(t) s 0, t 0 and h(t) /t-»ooas t-'oo,
and assuFiinj the uniform boundedness or a certain kind
1
of convergence of the sequence jL (h(|f|);x)j
(n=1,2,...) at a point x of continuity of the function
f(t), proved some converi^ence properties of the sequence
jL (f;x)j (n=1,2,...) to f(x) as n -* oo . This approach
does not require an explicit knowledge of the kind of
unboundedness of f(t), Kov/ever, v/hereas the use of a
bounding function gives rise to a v/iiole class of
ujibounded function.s for v/Lich the approximation takes
place, the use of a "W'-function h(t) requires computations
for the sequence {L (h(|f|);x) }for each f. Tne concept of
the bounding functions is certainly better than that
of the V/-functions since in any case h(|f|) is a
bounding function for the function f.
2_, \/hile developing a general theory of linear
combinations of linear positive oper-.tors, the author
[54] introduced a concept of local unsaturation of
linear positive operators on positive zero orders.
Besides linear combinations, this concept plays a very
useful role also in the theory of simulti.neous
approximation by linear positive oper .tors, linear
combinations of trieir iterates and the approximation of
unbjunded functions, Ilain results based on this concept
are described as follov/s-r
Let>(z) denote a class of non-nei^ative functions
F (t,x), (a ranging over an index set l) of two
variables t and x, \.'here each F (t,x) satisfies the a
follov/ing properties: (a) F (x,x) = 0, (b) for an arbitrary 6 > 0» F (t,x) has a positive lo\/er bound
a for all t satisfying | t-x | a 6 v/hile x retains a fixed
value, (c) keeping x fixed, P (t,x) is continuous at
t=x and (d) there exists a trcaisitive relation denoted
by the "greater tl.an" sign (> ) sue. t]iat c(,p £ I,a > P
2
imply
F^(t,x) (1) lim •• f,^\ = 0 (keeping x fixed)
t -. X ^p^^'^-'
and that given a positive nuxiber if there exists a
positive number M such that v/henever P. (t,x) g N
(keeping x fixed), there holds P (t,x) g MP (t,x), p ex
Let JL } (n=1,2,...) be a se'- uence of linear
positive operators such that for each a £ I and a
fixed X, the function P (t.x) £ D, the domain of
definition of the oijer-tor seouonce jL 1. Further
assume that for each a £ I there holds (2) lim L (P (t,x);x) = 0. \ / n^ a
n — CO
Let a,p £ I and a > p . There are the following
three possibilities (if necessary, restricting to
sub-sequences):
(i) L (F„;x) = o(L (F ;x))
(ii) L^/Fp;x) = 0(L^.^(P^;x)), and
(iii) L (F ;X) = o(L (P„;x))
as n — CO , Here the symbol 0 denotes a strict capital
order. The lollot/ing three theorems wore proved in [54]
i/hich cover the possibilities (i)-(iii).
TilSORSi I If a > p , tliere does not hold the relation
(3) L^(Fp;x) = o(L^(F^;x)) , (n -* oo )
provided the function P is uniformly bounded by P a
whenever P is unbounded (that is to say given an
arbitrary N > 0 there exists an k > 0 sucii that v/henever
P (t,x) S K, there holds F (y,x) g HF (t,x), and without
any condition of P is bounded. ..Iso (j) is false if
3
for some y > « > p, L ' Y''''' ^ ° ' n p'"" ' ( n--), without any further condition.
THEOREM II If L (P ;x) = e(-7--r) and L (P„;x) = 0( / v ) n^ ' ^cp(n)' n^ p 9(,n) '
where (p(n) is non-zero and tends to oo v;ith n, and
a > p. then for no y > a can there hold
(4) L (F ;x) = o(—7—r), (n -* 00 ).
TH' .REII III If for some a > p there holds
(5) \^'^^^^-) = o(L^(Pp;x)), (n - c- ).
Then for every |i < p ,
(6) LjPp;x) = o(LjF^;x)).
P^r convenience the functions F (a £ l) are called a
"positive zeros" and the subscripts a are called the
corresponding "zero-oraers". If P has a nigher zero-
ordor than F (i.e. a> p ), from the ^oint of view of r
approxima,tion it seems natural to expect that the
convergence of the sequence JL (p ;x)} to zero De faster
than that of the sequence jL (F ;x) j . Ile-.ce \/e say tnat
j L } is saturated at a(a £ l) if a higher oraer of
zero does not im ly a better approximation, that is to
say, for n O Y > o : ( Y £ I ) there holds
L (P ;x) = O(L (P ;x)), as n—00 , On the other hand if n^ Y n a ^" for some Y > a there holds L (P ;x) = 0 (L (P ;x)), v/e
' n^ Y ^ n^ a say that JL ! is unsaturated at a . '' ' n'
In case P (t,x) and F„(t,x) have comparable zero a p
orders of same pairty, i.e. F (t,x)
(7) lim -Tf (^ „ \ = P
where p is a positive number and further if given an
arbitrary positive II ttiere exists an II > 0 such that
F^(t,x) g II and Pp(t,x) g il respectively imply
P ( t , x ) g MF„( t , x ) and P „ ( t ; x ) S HP ( t , x ) , we v ; r i t e a p p a a ~ p . The fo l lov / ing theorem of [54] d e a l s v/ith t h e case
a ~ p.
TKE0REI4 IV I f a ~ P and {L j i s u n s a t u r a t e d a t p , t han
t h e r e h o l d s
F „ ( t , x ) L (F ;x )
^Q) 1^"^ F ( t , x ) = ^^"^ L (P =x) •
t - * x p ^ ' ^ n — oo n ^ p ' ^
A converse of theorem IV is also true. If for every
a ~ p the relation (o) holds and if there exists a Y > a
such that given an arbitrary N > 0 there exists an M > 0
such that F (t;x) g N implies P (t;x) g MP (t;x), then
JL I is unsaturated at p. This converse proposition is
not included in [54]» which, however, can be proved
rather easily.
The definition of unsaturati^-n of {L } at p implies
the existence of a Y > P such that L (F ;x) = 0
(L (p„;x)) as n -* oo, in this context it was remarked
in [54] that the functions F and F_ in theorem IV may
be replaced by more general functions f(t) and g(t)
which have the same order of zero at t=x, are positive
in a neighbourhood of this point (encluding the point x),
are bounded by a function P outside this neighbourhood where F is such that fL | are not saturated before YI
Y ' n' ' '
has a zero of an order higher than that of f or g and
has other properties of this notation,
An example of a class of the type Z ) is given by
the set of functions P = It-xI , a £ R , the positive
real line excluding zero where a > P has the usual
significance.
The theory of local unsaturation of positive linear
operators on positive zero orders (together with
5
theorems I-IV and the remark after theorem IV) provides
a great deal of insight into the study of asymptotic
approximation fcrnulae of Voi-onovskaja type (for tvj-ice
differentiable (at x) functions) and those containing
higher derivatives upto an even order. For instance a
necessary and sufficient condition that a sequence
jL } of linear positive operators possesses an asymptotic
formula for twice differentiable (at x) functions of a
certain growth (if unbounded) is that the sequence JL }
be unsaturated at a second order positive zero having
a similar growth. For a higher asymptotic expansion
conta.ining derivatives upto f (x) a similar necessary
and sufficient condition is local unsaturation at a th
2m ' oraer positive zero of an appropirate growth. Prom
this point of vie v various equivalent formulations of
xureckii's theorem ([10], p. 76) are all obvious.
Above discussion of local unsaturation is usefull
for having a macroscopic viev/ of various results obtained
in tills thesis. It will also help in obtaining various
equivalent foriaulations of tiie given results v/iiica may
be easier to deal \/itii a particular situti-ti-n,
5. Contents of the thesis.
Chapters 1-4 deal v;lth the deternination of classes
of unbounded functions, members of which may be a.pproxi-
mable v/ith the help of a given sequence of linear
positive operators.
Chapter 1 is an extension of Schurer's work [62],
Under the sane (or milder) hypothesis as in [62], we
obtain results v/hich are ap_.licable to functions having
an unboundedness of a higher order. These results can be
generalized for otlier test f-onctions forming a
6
Tchebychev system (extended, in case of asymptotic
formulae), To indicate some of the manipulations in
volved we also include, in this chapter, some results
using the test functions 1, sin t, cos t,... and 1, + ott +2at (' -V n^ e— , e- ,... (a > 0;,
Chapter 2 generalizes some of the results of Kliller
[48] Hiiller and V/alk [50], Eisenberg and Wood [14] and
Sclimid [56] etc, based on the concept of bounding
functions and the \/-functions. Besides giving an
independent treatment based on the concept of bounding
functions we also make a unified treatment which
combines the two tecliniques of the bounding and the
W-functions, This unified treatment generalizes several
results of Sclimid [56].
Chapter 3 makes use of certain estimates connected
with the theory of local unsaturation of linear positive
operators (of summation type) on positive zero orders
and envisages its application in determining some
classes of unbounded functions which admit of an
approximation by means of these operators. We limit
ourselves to the case of the Bernstein polynomials only.
It v/ould, however, be clear that the method introduced
has a general applicability,
Chapter 4 deals v/ith the approximation of unbounded
but integrable functions by means of sequences of linear
positive operators of integral type. \/e limit our
attention to the generalized Jackson operators, the
Vallee-Poussin integrals, the Gamma operators of Iluller
and the singular integrals of of Gauss-Weierstra.ss,
Nevertheless, the method introduced seems to be of a
general applicability,
7
CHAPTER 1.
METHOD OP TEST FUNCTIONS.
The test functions used in this chapter are
1, t, t ,...; 1, s m t, cos t, ... and e (a > 0,
m = 0, + 1, + 2,..,). Parallel formulations of various
results proved can be obtained in terms of other test
functions (e.g. those forming Tchebycheff and extended
Tchebycheff systems). Some other useful test functions
are {exp a-r^, a g 0|, Jexp a(s ), ex £ E, the real
1 s s
line! f ll, t"'', t"^,...| andje"/"* , a g OJ.The first
three of these have been used by the author [54] in
connection with several operator sequences generated by
special functions. The last one provides a deeper
understanding of approximation properties of the Gamma
operators of Miiller introduced in [45]•
1,1. Approximation of functions having at the most a
polynomial growth when the variable tends to + QQ.
1,1,1.Definitions and notations
Let X, X be two given subsets of R and let
L ( X ) , D(X) be two linear spaces of real (or complex)
valued functions defined on X and X respectively (the
scalar field being the set of real (or complex)
numbers). In the complex case it is assumed that if
f £ D(X) then also its complex conjugate f £ D(x), Let
U be an unbounded index set of positive real numbers
and let {L , n £ Uj be a class of linear operators
8
mapping D(X) into D(X), The class {L , n £ Uj is said
to be ultimately positive on a point set X (by definition
X S.X) if to each f £ D(X) and satisfying f(t) g 0 for
all t £ X, there exists a positive number n , say, such
that for all n £ U with n > n there holds L (f;x) > 0 o n^ ' ~
for all X £ X*. It would be convenient to call f(x)
respectively L (f;x)as the function and the operator
(L ) value of f at the point x, ..'ith n g 0, \ic defrne H „: class of all rea,l (or' cor.plex)valued functions n, X f(t) defined on X, to each of \/Lich there exist constants A,B > 0 such that |f(t)| < A+3(t)^ for all t £ X.
H ,,(x): class of all f £ H ,. \;hich have an extension m,X^ ' m,X
^ •
1 on R wnich is continuous at the point x.
H^''4(x) : class of all f £ H ^ which have an extension m,A m,X
f on R which is twice differentiable at the point x. In
the sequel f is used to denote both f and f. Similarly
the derivatives f'(x), f"(x),,., are denoted by
f'(x), f"(x),,.., respectively, Ii this context we
remark that the possibility of several extensions f
will lead to no ambiguity in our results.
<a,b> : some open interval (c,d) containing the closed
interval [a,b] .
H^ I <a,b> : class of all f £ H ^ which have an m,A ^ m,A
extension f on R which is twice differentiable at each
point of <a,b> with f" continuous at each point of
[a,b].
Let S be a subset of R. By K „(S) we denote m,A
the class of all f C H ^ such that for each x £ S, m,A
f £ H Y ( X ) ,
^y «Q,X' « Q , x ( - ) ' « Q,X^^) ' «^ Q,X<^'^> ^^^
H v ( S ) we d e n o t e t h e c l a s s e s of a l l f i i nc t i ons f such
9
(2) that for some positive m, f £ H ,,, H v(x), H^ v(x), /„>, m,X' m,X ' m,X H^ ^<a,b> and H ^(.S) respectively, m,A m,A (o\
Thus the classes H(x) and H^ ' (x) as defined
by Schurer [62] are identical with our classes H rjCx) (2) '
and E\ n(x) respectively, 2fR
1,1,2. Convergence of the operators L .
The following theorem is a generalization of a
theorem of Schurer [62], valid for functions of the
classes H_ Y(^) where m is a positive integer.
Schurer's theorem is related to the class H„ ^ (x) and
makes an improvement over a result of Bohman [5] and
Korovkin [30] (dealing with the class H„ -,([a,b]))
under the same premises. For m=1 we obtain the result
of Schurer. Unless otherwise clear form the context,
X will denote a fixed point.
THEOREM 1 Let m be a positive integer and m' an odd
positive integer such that m' < 2m, Let JL , n £ U)
be a class of linear operators defined on a common
domain D(X) (X S R) of functions into a domain
L ( X ) ( X S R ) of functions and ultimately positive on a
set 5f^X, Assuming that 1, t , t £ D(X) and writing
L (1;x) = 1 + a (x) n^ ' ^ n^ '
(1) L^(t'"';x) = x^' + p^(x)
T ('j_2m \ 2m / \ n^ ;x} = X + Yj (x) ,
where x £ X and n £ U, i f and only i f t he r e h o l d
(2 ) l i m a (x) = l im p (x ) = lim Y ( X ) = 0, \ / n^ ' ^n^ ' 'n^ ' ' n - » o o n - * c o n - » o o
10
then for each f £ D(x) r> H„ y(^) ^^ have
(3) lim Ljf;x) = f(x). n — oo
Further, let S ^ X be a compact set. Then for
each f £ D(X) r\ H„ yC^) relation (3) holds uniformly
in X £ S if and only if (2) hold uniformly in x £ S,
Note. Since in order to construct L (f;x), n £ U the
function values f(t) for t ^ X are not required, for
an arbitrary f £ H„ -^{x) rs D(X) one may expect the cm, A
convergence L (f;x) -» f(x) as n -» 00 only for the
points X £ 5?, the closure of X (b^ virtue of the
continuity of f). Tnus, at first sight it may seem
strange that in the formulation of above theorem we do
not include the hypothesis that x be a point or a
cluster point of X. In fact, such an explicit inclusion
would be redundant as it is already implicit in the
relations (1) and (2). To prove this assume that (1)
and (2) hold without x £ X. We show a contradiction. Let
p(t) be the polynomial as defined in (4) below. We have
shown in the sequel that for an arbitrary 6 > 0, p(t)
has a positive lower bound, say m., on the set
jt : I t-x I g 6] , if X ) X, there exists a 6 > 0 such
that (x-6, X +6) r X = 0. Hence for all t £ X there
holds the inequality
0 g 1 g m""" p(t).
By the ultimate positivity of JL ,n £ U] at the point
X it follows that for all sufficiently large n £ U
we have
0 g Ljl;x) g m '' L^(p(t);x).
11
From this, using (l) and (2), we reach at
lim L^(l;x) = 0, n -> 00
which IS a contradiction,
A similar remark would be seen to be applicable
m all further results on the basic convergence of
operator values,
Proof of theorem 1. In both the assertions, the
necessity part of the theorem is trivial. To prove that
the conditions are suflicient consider the polynomial
p(t) defined by
/ , \ / i_ . .2m _ , m' 2m-m' / , , \ 2m (4) P('t) = in "t - 2mt X + (2m-m')x .
By Descartes' rule of signs it follo\7S that if x 4= 0
then p(t) has at the most two real zeros, multiple
zeros counted after their multi^l"cities and tnat they
have the same sign. Clearly t=x is a zero of p(t), ,/hen
x=0, t=0 is the only zero of p(t) and we have p(t) > 0
if t 4= 0, If X + 0, v;e have p'(x) = 0 so that t=x is a
double zero and tnerefore p(t) has no other zero and
since p(t) IS of an even degree we nave p(t) > 0 if
t 4= X, hence for every 5 > 0, p(t) has a positive lower
bound on the set jt : | t-x| > b],
Since the op'--ra,tors L , (n £ U), are linear
it IS sufficient to prove the result for real valued
functions f £ D(X) rs H_ v(^)* ' hen the functions ^m, A
L (f) are also real valued. (This remark snail 'oe made n '
use of m all suosequent results). By the continuity
of f(t) at t=x, given an arbitrary e > 0, there exists
a 6 > 0 such that
|f(t) - f(x) 1 < e , (lt-x|< 6, t C X).
12
Also there exists a jjositive real number A such that
|f(t) - f(x) I < A p(t), ([b-x|g 6 , t £ X),
3y non-negativity of p(t), therefore
|f(t) - f(x)| < e + A p(t), (t £ X).
The functions c + A p(t) + (f(t) - f(x)) belong to
D(X) n, H„ y(^)« Hence by the ultimate positivity of
|L , n £ Uj there exists a number N > 0 such that
L ( c + A p(t) + (f(t) - f(x)); X ) g 0, n > N^,
By the linearity of L we then have
|L^(f;x) - f(x)| g 1 L (1.;x) - l||f(x)|
+ e L^(l;x) + A L^(p(t);x),
Using (l) and (2) we can find an Np > 0 such that
|L^(1;X) - I| 1 f(x)| < z
L^(l;x) < 2 I n > N^ .
and A L^(p(t);x) < e
Let N = max JN , N j. Then
|L^(f;x) - f(x)| < 4£ , n > N,
By the arbitrariness of e > 0 it follows that
lim L^(f;x) = f(x), n — oo
proving the first assertion,
To prove the second assertion, assume on the
contrary that under the given assumptions (3) does not
hold uniformly in x £ S, Then given an arbitrary c > 0
13
there exists a sequence jn, , n, £ Uj of numbers and a
sequence jx, , x, £ Sjof points satisfying n, - oo ,
k — oo and lim x, =x for some x £ S such that k - oo
(5) |L (f;xj ) - f(x^)l > e , k=1,2,.,. .
On the domain D(x) of functions define at x
a sequence {M, j (k=1,2,,..) of operators by the relation
(6) M^(f;x) = L (r;xj^).
Then, writing
Mj^(l;x) E 1 + %(xj^) = 1 + cej (x)
(7) ^it'^'ix) = xj' + p^(x^) = x" ' + p'(x)
Mj (t ^;x) = x^ + Yj, (xj ) = X ™ + Yj (x) ,
(the primes are just a convenient notation and do not
represent derivatives) since a , p , Y "* uniformly
on S as n -• oo and since x, — x as k — oo , it follows
that
(8) lim aj[.(x) = lim P^(x) = lim y^M = 0. k-»oo k-'oo k-»oo
By the first part of the theorem and (6) it follows
that
(9) lim Mj^(f;x) = f(x). k -* oo
Also since f(t) is continuous at t=x, lira f(x, ) 1 k k — oo
= f(x). Hence there exists a natural number N, such
that
(10) l n, '""k " ^ ""k l < e , k > Nj,
14
Since (l0) contradicts (5), the second assertion of the
theorem follows. This completes the proof of the theorem,
Remark 1, In the statement of theorem 1, (2) can
equivalently be replaced by
(11) lim {L^(l;x) - If = lim L^( (t-x)^°'3x)- 0, n -* OO n -* oo
(assuming that (t-x) £ D(X)), To prove this fact we
replace the polynomial p(t) by the polynomial (t-x)
in the proof of theorem 1 and argue as before.
Remark 2. In the light of theorem 1 we point out to
an improvement on the following result of Hsu [17] •
THEOREM I Let JL j (n=1,2,,,,) be a sequence of linear
operators such that for all large n and every f(t)
belonging to the domain of definition of JL j and
non-negative for -oo < t < oo we have L (f;x) g 0 for
all X £ [-1,1]. Let {a j be a sequence of real numbers
increasing to + oo with n(a 4= O) and let the following
limit relation
(12) lim L ((a t)^; a"''x) = x^ \ / n ^ n ' ' n ^ n -> oo
exist and hold uniformly for all values of x in every
finite interval, where k = 0,1,2,m,m+1,m+2; and m is a
non-negative even integer. Then for every function f(t)
defined and continuous on (-00,00) and satisfying the
condition
(15) f ( t ) = o d t D , ( t ->±00)
we have the limit relation
(14) lim L^(f(a^t); a~^x) = f(x), n -• 00
15
(_oo < X < oo). Moreover, this relation holds uniformly
on every finite interval of x,
Our improvement on this theorem is as follows:
if m=2, condition (12) is superfluous for k=3,4. If
m g 4, condition (12) is superfluous for k=1,m+1,m+2,
Also in the latter case condition (l2) for k=1 can be
replaced by (12) for any odd positive integer k < m,
To prove this assertion, consider the sequence
(L*j(n=1,2,,..) of operators defined by
(15) L* (f;x) = L (f(a t); a~''x), n=1,2,... \ ^ I n ^ ' ^ n ^ ^ n ^ ' n ^ ' ' '
where the L are as in the above theorem. Asstime that n
(12) holds uniformly for all values of x in every finite
interval for k=0, m', m, where m' is an odd positive
integer less than m. (in particular we may take m'=l),
Replacing L by L* and 2m by m in theorem 1, the result
of Hsu follows,
Remark 3. If the class {L , n £ Uj of operators is
restricted to the non-negative real axis (i.e. to
construct L (f;x), (n £ U), the values f(t) for
t £ (- °°, 0) are not required or equivalently
X :^R E [0, 00)), then theorem 1 can be generalized
to the following,
THEOREM 2. Let m, m' be two pooitive numbers with
m' < m. Let JL , n £ UJ be a class of linear operators
defined on a common domain D(X) ( X ^ R ) of functions
into a domain D ( X ) ( X C R ) of functions and ultimately
positive on a set X X , Assuming that 1,t ,t £ D(Xj
and writing
16
L (l;x) = 1 + a (x) n^ n^ '
(16) L^(t"'';x) = x""' + p^(x)
L (t ;x) = X + Y (x), n^ ' 'n '
where x £ X and n £ U, if and only if there hold
(17) lim a^(x) = lim p^(x) = lim Y^CX) = 0, n->oo n — oo n-*oo
then we have for each f £ D(x) n. H ^(x) ^ m,X '
(18) lim L^(f;x) = f(x), n — oo
Further, let S ^ X be a compact set. Then for
each f £ D(X) ^ H ^(S) relation (l8) holds uniformly m, A
in x £ S if ond only if (17) hold uniformly in x £ S.
Proof. Consider the function
I ,\ ,,m ,m' m-m' / • \ m q(t) = m't - mt X + (m-m')x ,
for t g 0 and for a fixed x g 0. Then t=x is a zero
of q(t). If y g 0 (y 4 x) would we another zero of q(t), m' m'
then u = X ,y would be two distinet zeros of the
function
/ N , m/m' m-m' / . \ m v(uj = m'u - mux + (m-m'Jx ,
Applying Rolle's theorem, there would then exist a m' m'
positive number ^ lying bet'..een x and y , such that
• ^ - 1 ,/„N „ m' m-m' ^
v ( U = ' n ^ - m x = 0 ,
m' Clearly this implies that ^ = x , vAich is a
contradiction. Hence the only non-negative real zero
of q(t) is t=x. It is clear that because m > m' we have
q(t) > 0 if t 4= X and that q(t) has a positive lov/er
bound on the set ft :It-xI g 6} for each 6 > 0,
17
Replacing the polynomial p(t) by the function
q(t) in the proof of theorem 1 and proceeding analogously
we can complete the proof of theorem 2,
1,1,3. Asymptotic formulae for twice differentiable
functions.
Next we consider the existence of an asymptotic
formula giving rate of convergence of L (f;x) to f(,x)
for twice differentiable functions. In connection with
Theorem 1 of [62], Schurer remarked that the
corresponding theorem of Bohman-Korovkin, although 2
utilizing the unbounded functions t and t , gives the
convergence only for bounded functions. Interestingly
enough, his next result. Theorem 2 [62] utilizes the
function (t-x) for some positive integer m, which
has the unboundedness of the order t , 11 | - oo and
yet produces a result only for f(t) = 0(t ), |tj -* <»,
With the help of the following theorem, with no extra
assumptions we can obtain the result for f(t) = 0(t ),
I t I — oo,
THEOREM 1, Let m > 2 be a positive number. Let
JL , n £ uj be a class of linear operators defined
on a common domain D(X) ( X ^ R) of functions into a
domain L ( X ) ( X ' ^ R ) of functions and ultimately positive
on a set Xc^x. Assume that x £ X and that the functions
1,t,t^, It-xl" £ D(X), If and only if there hold
18
and
^o(x) ^ L ( l ; x ) = 1 + —7—r + o(—7—v)
n^ 9 (n) > C n ) ' '? (x )
(1 ) L ( t ; x ) = X + "'/ ^ + o( / \ )
2 2 2 "' ' 1 L ( t ; x ) = X + 7—r + o(—7—v)
(2) L^(|t-xr;x) = o ( ^ ) ,
as n -> 00, v/here (p(n) 4= 0, q)(n) -> 00 as n -* oo,
t h e n f o r a l l f £ D ( X ) r H^^^(x) t h e r e e x i s t s t h e m,X
asymptotic relation
(3) L^(f;x) -f(x) = - ^ [f(.),^(x)
+ f'(x){T/x) - xw^(x)j + ^ ^ {?2(x)
-2x?^(x)+x2l'^(x)j] + o ( ^ ^ ) , n -' 00 .
Further, if [a,b] is an interval ([a,b]^X)
such that for each x £ [a,b], the function |t-x| £ D(X)
then for all f £ L(X) rs H^^^<a,b>, the formula (3) ' m,X ' ' ^^'
holds uniformly in x £ [a,b] if and only if (1) and (2)
hold uniformly in x £ [a,b] , provided that the function
?2(x) - 2x'?.(x) + X V (x) is bounded for x £ [a,b], Remark 1. If x is not a cluster point of X, for an
(2) f £ D(X) r\ VL- Y(X) we can have many extensions of f
which will assign arbitrary values to f'(x) and f"(x)
(the value f(x) being an exception if x £ X without
being a limit point of X). However, this fact does not
lead to a contradiction in relation (3)» since in such
a case 'I'.(x) - x F (x) and 5' (x) - 2x1'(x) + x 'If (x) are
automatically both zero. To prove this v/e observe that,
in this case, there exist positive constantsA and B
19
such
and
that for all t £ X
t-x < ii t-x 1
/-J. ^ 2 , -o , im (t-xj < B t-x
Using (2) we have then
L^(t-x;x) = o( ; j | ^ ) , (n -> 00 ) n- ^fsf\
2 1 and L^(( t -x) ;x) = o(rnn") »(n -^ °° ) ^
from which the assertion follows.
A similar remark is applicable in all further
results on asymptotic formulae and estimates and
therefore any more reference to this point will be
omitted,
Remark 2. If v;e assume that the function 2
l'p(x) - 2x?^(x) + X ? (x) is unbounded on [a,b] but
that (1) and (2) hold uniformly in x £ [a,b] then an
application of Holder's inequality shows that V (x) 0
must be unbounded on [a,b] . Consequently for all
sufficiently large n the function L (l;x) is unbounded
and therefore discontinuous on [a,b]. Thus in a
practical approximation method the unboundedness of
^^(x) - 2x 1' (x) + X 1' (x) is unlikely to arise,
Remark 3' In the second part of tneorem 1, dealing
with the uniformity of relation (3) on an interval
[a,b], we have made the assumption (in the definition (2)
of the class H -5.<a,b>) that the seG>.nd derivative f" m, A
be continuou" ,",t each point of the interv::l ^a,b].
However, Suzuki [74] (Theorem B, p, 451) in his
formulation (wnich is given v/ithout a proof) of Mamedov
[36] and Scnurer's tneorem (Theorem 2, [o2]) does not
20
assume this continuity. His statement, which we
reproduce exo.ctly, is as follows: (The sequence {L j
of linear positive operators, occuring in the following
theorem, is assumed to be mapping C[a,b] into C [a,b],
where [a,b] is a finite interval).
THE OH EIM I Assume that the sequence of linear positive
operators {L (f;x)j has the property that
L^(l;x) = 1 , X £ [a,b] ,
L (t;x) = X + —7—V + o(—7—v)f uniformly over n^ ' 9(n) ^cp(n)" ^
' o(x) [a,b],
L^(t ;x) = X + /^\ + o(-7^). uniformly over
[a,b],
If there exists a positive integer m ( > 1) such that
Lj^((t-x) ";x) = °('^(7T)> uniformly over [a,bj ,
then for each f(x) £ J [a,b ], we have
2f'(x)Y (x)+f"(x){ Y (x)-2xT. (x) j L (f;x)-f(x)= ^-^-7-.; ^ + o(-7^) n^ ' ^ ^ ' 29(n) ^(n)^
uniformly on [a ,b ] , a < a < b < b, where C •^[a,b]
is the set of all real functions f(x) of which the
second derivatives f"(x) exist in [a,b] and are bounded.
In the same paper Suzuki used theorem I to prove
Proposition 1, p. 432; Proposition 4> P« 434;
Corollary 1, p. 437; Corollary 2, p. 437 and Theorem 1,
p. 438 (in which actually Poposition 1 is used).
In Theorem 1 the operators L considered are such that n
they preserve linear functions and '{'p(x) is boundea,
twice continuously differentiable and not equal to zero
21
on (a,b).
In fact the statement of tneorem I is not true.
We give an example of a sequence L j of linear positive
operators mapping C[-2,2] into C[-2,2]preserving
linear functions and satisfying the assumptions of
theorem I witn ^p(x) bounded, tv/ice continuously
differentiable and not equal to zero on (-2,2) for
which the asymptotic formula given by theorem I does
not hold unii .irnly on [-1,1] for a function
f £ C^2) [_2,2].
Let jL j (n=1,2,...) be the Sequence of linear
positive operators, miapping C[-2,2] into C[-2,2J,
defined in the follov/ing way:
i[„, Jisdi^Js^^ , ,(,. vSujUi!; (2n+ JTt
L (f;: n^
(2n+ -l)::
1 < |x
• )
< 2
^ [i(x+ (2n+ ^)n
1,2,... , We have
r(x-
L^(l;x) = 1, x £ [-2,2] ,
L v"fc;x) n^ ' e [-2,2] ,
(2n+ •^)n
T rx2 N 2 y (x) . 1 -, L^(t ;x) = X + — ^ + o(-2) ,
4rt "n n
uniformly over [-2,2j, where
f |X| S 1,
'P(x) = 1, i x| g 1
Also 1-(| x!-l)^, 1 < Ixl g 2 .
. ((t-x)'^'^;x) = o ( ^ ) , uniformly over [-2,2] , n j^^
22
where m > 1 is an arbitrary positive integer.
Assuming that theorem I
f £ c'-^^[-2,2] we would have
Assuming that theorem I is true, for each (2),
1 t^ \ ff -S ^(x)f"(x) ^ / I N
8-11 n n
uniformly on [a^,b.] where -2 < a < b. < 2, Taking
a =-1, b.=1, for an arbitrary e > 0 we can then find
an integer N > 0 such that for all n > N. and x£ [-1,1]
there holds
|m2{f(x-t^) + f(x-^) - 2f(x)j - f"(x)| < e
-1 where m = (2n+T-)7i and f is a fixed element of (2) 1 C "^[-2,2], Choosing x=0 and —, in turn, we have
I 2 f{^) + f(-^) - 2f(o)j - f"(o)| < e , and
f(^) + f(o) - 2f(-i)j - f"(^)| < £ ,
whenever n > N., It follows from these two inequalities
that
|m2|f(|) + 3f(o) - 3f(^) - fi-i)]-{f"ii) - f(o)}|<2e
whenever n > N., By L'Hospital's rule
lim m2(f(|) + 3f(o) - 3f(^) - f(-i)! m — 00
= lim f (2f(^) - 3f'(-) + f'(-^)} 2 ' ^m' ^m' ^ m'' m -* 00
= lim m!f'(|) - f'(o)j - lim -^{f' ( )-f' (o)j m -• 00 m -• 00
+ lim f (f(-l) - f'(o)j m -* 00
= 2f"(o) - I f"(o) - f"(o) = 0.
23
Hence there exists a positive integer Np such that
|m^{f(^) + 3f(o) - 3f(-) - f(--)j| < £
for all n > Np. Thus for all n > N ,Np we have
f "( \ ) - f"(o)| < 3e . (2n+2)Ti
Consider the following choice of the function f.
0, if x=0,
f(x) =
x^sin -, if X £ [-2,2] and x 4= 0.
It is easy to check that f(x) £ C^^[-2,2] and that on
the interval [-2,2] its second derivative f"(x) exists
and is given by
0 , if x=0,
f"(x) = j (12x^-1)sin - - 6x cos -,if x £[-2,2],x40.
Clearly we can choose a positive integer N, such that
for all n > N, we have 3
|12( ——)^sin(2n-4)7i - —^—cos(2n-4)Ti| < e. (2n-t )7i (2n4^)K
Thus for all n > N , Np, N, we have
I 1
|sin (2n+2)7i| < 4e •
As e > 0 is arbitrary, this gives us a desired
contradiction.
Proof of theorem 1. Writing
24
(4) f(t) = f(x) + (t-x)f'(x) + % ^ f"(x)
+ {(t-x)2 + It-xrjh^(t)
and putting h (x) = 0, the continuity of h (t) at t=x DC 3C
fellows. Thus given an arbitrary £ >0, there exists
a 6 > 0 such that (5) |h (t)I < e , for all t £ X with jt-xj < b .
Since there exist positive constants A and B such that
|f(t)| < A + Bltl"^ for ail t £ X, it follows from (4)
that there exists a positive constant M such that
(6) Ih (t)I < M, for all t £ X with |t-x| g b .
Obviously we can choose M so large that both of the
inequalities
(7) l\(t)| < M, and
ra\ lu ^ ^ I / Mlt-xT"^ (8) h (t) < e + —' hi— ^ ' ' x^ ^ ' ,m-2
o
are valid for all t £ X.
By linearity and the positivity of L , using
(7) and (8) we have the inequality
(9) |L^(f;x)-f(x)L^(l;x)-f'(x){L^(t;x)-xL^(l;x)
- ^^{L^(tSx)-2xL^(t;x)+x^L^(l;;
n'
;x;
= lLj{(t-x)2+|t-xrjh^(t);x)|
S e L^((t-x)^x)+M(l4^j;^)L^(|t-x|"';x),
6
valid for all sufficiently large n.
It follows from (I) that, for all sufficiently
large n,
25
"f^M ^ ^ ^^ ^^ ^^^ ^T^(x)-xy^(x)
9 (10) |r(,){-°_^+1 -L^(l;x)j+f.(x)i ^
-L^(t;x)+xL^(l;x)j+ n
^„(^) Y2(^)-2x^-,(x)+x^?o(x) 2 5 ^(^
- L (t ;x) + 2 x L (t;x) - x L (l;x)j| g -f-y n^ ' n^ ^ n^ ' 'I 9(n)
and also that
(11) L^((t-x)2;x) g ^ ( )
where K is a positive number independent of n and e,
Again by (2) for all sufficiently large n
(12) M(1 . - ^ ) L ^ ( l t - x r ; x ) g ^ .
Combining above inequalities, it follows that for all
sufficiently large n
(13) |9(n)iL^(f;x) - f(x) - (7y[i"(x)y (x)
+ f'(x)fY^(x) - xV^(x)j + ^ f"(x){'?2(x)-2x? (x)
+ x^Y^(x)j]j| g e (2+K).
Since e > 0 is arbitrary (3) follows.
For the uniform convergence part, by a mean value
theorem
(14) f(t)-f(x)-f'(x)(t-x) = -^^=1^ f"(0
for some E, lying between t and x, if f" exists at all (2) points between t and x. If f £ H /. <a,b>, applying m, A
(Lemma 1, p,12, Korovkin [30]) to the function f", we
find that given an arbitrary e > 0, there exists a 6 > 0,
independent of x, such that
26
(15) | f " ( t ) - f " ( x : S £
f o r a l l t £ X such t h a t | t - x | < 6 , x £ [ a , b ] . Hence
from (4) we have t h e i n e q u a l i t y
(16) 2 | h ^ ( t ) | j l + | t - x r - 2 j = | f " ( 0 - f " ( x ) | § £
for all t £ X and satisfying |t-x| < 6 ,x £ [a,b].
It also follows that there exists a constant M such that
(7) and (8) hold for all x £ [a,b]. By the boundedness
of the function Y (x)-2xy (x) + x 'F (x), K in (II) can
also be chosen to be independent of x. In this way the
right hand side of (13) becomes independent of x and
the uniform convergence follows.
As the necessity parts in both the assertions of
the theorem are trivial this completes the proof of the
theorem.
COROLLARY 1,1 Let m^ and mp be two positive numbers
where m > 2, Let (L , n £ UJ be a class of linear
operators defined on a common domain D(X) (X^=.R) of
functions into a domain D(x) (X & R ) of functions and
ultimately positive on a set X £^X, Assuming that ~ 2 I i™1 I * imi I |m2 . ,
X £ X and 1,t,t , |t-x| and |t-x| |t| £ D ( X ) , if
and only if (1) together with the conditions
m . (17) L (|t-x| x ) = o{-i-v) ^ ' n^' I ' ' 9(n) and
(18) L^(it-x| ^ t ) 2;x) = o ( ^ )
hold as n ^ 00, where 9(n) 4= 0, (p(n) -* 00 as n -* 00 , then
the asymptotic relation (3) holds for each
f £ D(X) r. H^2) u). m^+mp.X ^ ^^
Further, assuming that [a,b]^X,|t-x| and
27
|t-x|"^'^ h i ^ G D ( X ) f o r a l l x £ [ a , b ] and t h a t t h e p
function T„(x)--2x'l'. (x) + x '? (x) is bounded on [a,b], (?) °
for each f £ D(x) r, E^ i ^<a.,-b>, (3) holds uniformly m.+mp,A
in X £ [a,b] if and only (l), (17) and (18) hold
uniformly in x £ [a,b],
Proof. The necessity parts in both the assertions of
the corollary are trivially true. In order to prove the
sufficiency parts, we observe that two positive constants
A and B can be found such that for all real values of t
the following inequality holds
m.+m m m. m (19) | t - x | g A | t - x | + B | t - x | | t | ,
m +m2 f o r each x £ [ a , b ] . Hence, assuming t h a t | t - x |
£ D ( X ) , f o r a l l n s u f f i c i e n t l y l a r g e
m +mp m (20) L ^ ( l t - x l ; x ) g A L ^ ( | t - x | ' ; x )
m mp + B L ( I t -xl It l ;x)
by (17) and (I8), Thus, in case (17) and (I8) hold
uniformly in x £ [a,b], the o-term in (20) holds
uniformly in x £ [a,b]. Now with m=m^+m^, theorem 1 is
applicable and the sufficiency parts of the corollary
follow, m.+mp
In case | t-x | ^ I^(x), we go back to the proof
of theorem 1 and replace the function |t-x| everywhere
by the function A|t-x| + B|t-x| |t| and proceed
analogously. This completes the proof of the corollary.
COROLLARY 1.2 Let m g 4 Ie an even positive integer,
Let JL , n £ uj be a class of linear operators defined
28
on a common domain D ( X ) ( X £ R ) of functions into a
domain D ( X ) ( X ^ R ) of functions and ultimately positive r^ '- r^ 2 3 4
on a set X X . Assume that x £ X and 1,t,t ,t ,t , ^m-4^^m-3^^m-2^^m-1 ^^^ ^m ^ ^^^^^ ^^^ ^^^^^^ ^^^^^ ^^^
Tp(x) be some functions of x. Then a necessary and
sufficient condition for (3) to hold for each (2)
f £ L(X) r> H^ ^(x) is that it holds for the functions
i,u,,..,u ancL u ,,.., X .
Further, assuming that [a,bj^X and that the 2
function ?p(x)-2xT (x) + x ¥ (x) is bounded on [a,b],
for each f £ D(X) r^ H^^2<a,b>, (3) holds uniformly in m,A
X £ [a,b] if and only if it holds so for the functions 1 + +4 ^ +m-4 .m
Proof. The necessity parts in both the assertions of the
corollary are trivial. To prove the sufficiency parts, 2
we note that for f(t) = 1,t,t , the relation (3) is
identical with the respective relations in (1). Now,
since f,f',f" occur linearly in (3), choosing m.,=4 and
mp=m-4 we find that (17) and (I8) are satisfied. Now
corollary 1.1 is applicable, completining the proof of
the corollary 1,2,
COROLLARY 1,3 Let m g 4 te an arbitrary positive
integer (i.e. not necessarily even). Let {L ,n £ Uj be
a class of linear operators defined on a common domain
D(X) ( X ^ R " ^ ) of functions into a domain D(X)(X^R'*')
of functions and ultimately positive on a set X.^X,
Assume that x £ X and 1,t,,.., t and t ,...,t £ D ( X ) ,
Let Y (X),'? (x) and Tp(x) be some functions of x. Then
a necessary and sufficient condition for (3) to hold
for each f £ D(X) H^^|(x) is that if holds for the ^ X • -IX x4 •, xm-4 xin functions 1,t,...,t and t ,...,t .
29
Proof. Choose m.=4, mp=m-4. Then (19) with |t| mp '
replaced by t ' is valid for all t g 0. Rest of the
proof follows along the lines of the proofs of the
corollary 1,1 and the corollary 1,2,
Remark 1. In the special case when ?p(x)-2x'i' (x) 2
+ X f (x) = 0, relation (2) is already fulfilled with
m=2. The second derivative f"(x) then does not occur
in (3) and this formula reduces to
(21) L^(f;x) - f(x) = ^[f(x)^o(x)
+ f'(x){¥,,(x) - xf^(x)j] + o ( - ^ ) ,
In fact the existence of f"(x) is not necessary in this
case and it is sufficient to assume that, with an
extension of f, f'(t) exists for t belonging to some
neighbourhood, say (c,d), of the point x and that the
first divided differences of f'(t), at the point x, are
uniformly bounded for all sufficiently small step-
lengths h, say |h| < 6 where 6 > 0. To prove this we
proceed as follows. By the assumptions on f'(t), for a
sufficiently small & > 0, there exists a constant M > 0
such that
if'(x+h) - f'(x)i I h 1 ^ ^ ^ '
whenever |h| < 6 (in this and the following step we are
working with an extension of f and so we need not
specialize x+h,t and ^ to belong to X), By a mean value
theorem if S > 0 is sufficiently small and I t-x | < 6,
we have
f(t)-f(x)-(t-x)f'(x) = (t-x)(f'(0-f'(x))
30
where ^ is a point lying between t and x and therefore
|(t-x)(f'(0-f(x))| g(t-x)2|^l^lM^Iil|.
Now we can specialize t to belong to X and have
(22) |f(t)-f(x)-(t-x)f'(x)I g M(t-x)^
wheij.ever |t-x| < 6 and t £ X, Rest of the proof follows
on the lines of the proof of theorem 1,
Formulae of type (21) occur in certain
perturbations of the Baskakov sequences recently studied
by Sikkema [70].
Remark 2 In the case when (2) or similar conditions
do not hold or are not known to hold, the following [l 1 result may, nevertheless be applicable. By H^ ^(x) m, A
we denote the subclass of H „(x) consisting of the m,X
functions f which, with an extension, possess a first
derivative at each point in a neighbourhood say (c,d)
of the point x such that at the point x all the first
divided differences of f'(t) with sufficiently small
step-lengths h, say |h| < 6 where 6 > 0, are uniformly
bounded by M„(x), say, i.e. |f'(t)-f'(x)| < M^(x) |t-x|
for all t satisfying 0 < |t-x| < b for a sufficiently
small 6 > 0. If for each x £ [a,b], f £ H'-''-!(X) and if '- ' •'' m,X^ '
the set {M„(X), X £ [a,b]j of numbers is bounded, say by M > 0, then we write f £ H- -! [a,b]. It is clear
that such an M exists if for same 6 > 0, f' £ Lip,,
on the interval [a-6,b+6] ; also that it is necessary
to have f £ Lip. 1 on the interval [a,b]. f
THEOREM 2. Let m g 2 be an even integer. Let [L ,n £ Uj
31
be a class of linear operators defined on a common
domain D ( X ) ( X S - R ) of functions into a domain D(JJ)(X^R)
of functions and ultimately positive on a set S!^X.
Assume that x £ X and 1,t,t ,t ~ ,t ~ and t £ D(X),
Then for each f £ D(x) r^ H'-''J(X) there holds
(23) L^(f;x) - f(x) = O ( ^ ) ,
as n -* oo, where (p(n) 4= 0, q)(n) - ooas n — oo, if and
only if it holds for the functions 1,t,t ,t ,t
and t .
Further, assiuning that [a,b] X , for each
f £ D(X) HL''^[a,b], (23) holds uniformly in x £ [a,b] m,A 2 _._p
if and only if it holds so for the functions 1,t,t ,t , ,m-1 , ,m t and t .
/ +
Also, in the case when X,X R , the above asseri
assertions are true even when m is an odd integer g 2.
A proof of theorem 2 can be given in a way similar
to that of the proof of theorem 1 where instead of
using (4) we start with the inequality (22) of remark 1.
We omit the details.
1.1.4 A class of linear positive operator sequences
In tnis section we apply the results of sections
1.1.2-1.1.3 to the case of the Baskakov-eequences of
linear positive operators and thereby extend an earlier
study of them made by Schurer [62]. First we give a
brief resume of the results of Schurer.
Schurer [62] considered the sequence jL j(n=1,2,...)
of operators defined by
^ cp' '(x) X
(1) L (f;x) = s (-i)^-^Hn ^(zh\)' n k=0 k! '^^(K)
32
THEOREM III Let I9 (x)j be such that we have the
special case
x(n) = n, 'i'(n,x) = n,
'?(m(n),x) = m(n), m(n) = n+c,
(for all sufficiently large values of n), where c is an
integer indepenuent of n and a, (x) are independent XV, n
of k, say, a (x). If the a (x) possess the property
that at a fixed point x £ [o,b]
6) «n -) = - "(7(^)
v/here T(n) has the prop rties: I) T(n) 4= 0; 2)
lim T(n) = 00; 3) T(n) = o(n), n -* 00 , then we have n - oc , ,
for f £ Up ^+(x)
(7) L„(f,x) - r(,) . -P(' j;;(-) . o ( ^ ) .
Before obtaining convergence thev,rems for classes (2)
larger than the classes Kp _^(x) and Kp 4.(x) of
functions, we note that no particular purpose is served
by taking n to be an integer in the definiti -n of the
operators L . We assume, therefore, that n £ U where
U is an unbounded set of positive real numbers. With
this stipulation the numbers m(n) also need not be
integers. Further, it v/ould be sufficient to assume
that the properties i)-iii) hold for all sufi'iently
large values of n.
The follo\/ing theorem extends the result of
theorem 1 of Scaurer to the classes H p+(x) and
Ji„ -n+ -fl j wituviut any extra assumption.
ThliOREM 1 Let f £ H p+(x), x £ [o,b]. Then for the ' ' ' ' i.6 f -iL
35
where jx(n) } (n=1,2,...) is a sequence of positive
numbers increasing to infinity with n and the sequences
W (x)j(n=1,2,...) of functions possesses the following
properties on an interval [o,b] (o < b < °o) s
i) V^{°) = 1
ii) 9 (x) is infinitely differentiable and
(-l)^9^^\x) g 0, (k=0,1,2,...);
iii) there exists a positive integer m(n), not
depending on k, such that
- 9i")(x) = .(n,x)cpif;))(x) il.a,^,(x)},
k=1,2,..., where
iii) a, (x) converges to zero uniformly in k when
n -* oo, and
iii)p 'i'(n,x) satisfies the following properties:
(2) li, Ii^4- =1, and n -• oo A\ /
v\ -, • 'y(m(n) .X,
3) lim —^—^Y^—' n -* OO
Here and in theorems I-III it is assumed that f
belongs to a certain class of functions for which(l)
for n=1,2,... is meaningful. This depends on the nature
of the functions 9 (x).
Remark. It is known that a function 9(x) satisfying
(_l) (pV -'(x) g 0 (k=0,1 ,2,...; X £ [o,b]) has an
analytic continuation for |x-b| g b. Hence for 0 g a g b
the series
00 , vk ( k ) / N k „ (-1) 9'- aa)x
k' k=0
33
has a radius of convergence > a. It follows that for
m=U,1,2,..., the series
°° / ^ \k (k) / N k, m (-1 ) 9 (x) X k
k' k=0 ^•
is convergent for each x £ [o,b] . As a Cv.nsequence, if
f(t) is a function bounded on each bounded subset of R
and satisfies f(t) = 0(t"), (t — oo), for some m > 0,
then (l) for n=1,2,... is meaningful for sucn a function.
Schurer proved the follov/ing tneorems l-III:
THEOREM I If f £ H^ j +(x) and if the sequence 9 (x)
satisfies the conditions (i)-(iii) then the sequence
{L (f;X)}(n=1,2,...) defined m (1) converges to f(x)
when n — 00, if^ moreover, m (iii)-, a, (x) converges I iC, n
to zero uniformly in x on [o,b] and if the relations
(2) and (3) hold uniformly m x on [o,b], then the
sequence JL (f;x)j (n=1,2,...) converges uniformly on
[o,b] to f(x), assuming f(x) £ Hp j +(x) (O g x g b).
THEOHEIi II Let {9 (x)j be such that we have the special
case
x(n) = n, 1'(n,x) = n
'i'(m(n),x) = m(n), m(n) = n+c,
(for all sufficiently large values of n), i/htre c is an
integer independent of n and a, (x) are independent rC I n
of k, say a (x). If the a (x) possess the property that
at a fixed point x £ [o,b]
(4) a (x) = ^ + o(l) ' n^ ' n ^n^
t hen we have l o r f £ lip n+(x)
(5 ) L ( f ; x ) - f ( x ) = 2 x p ( x ) f ' ( x ) 4 - f " ( x ) ( x + c x ' ) ^ _^^1^ \ ^ / j ^ \ ; / \ / 2n ^n^
34
TIIEuREM III Let {9 (x) j be such that we have the
special case
x(n) = n, 'i'(n,x) = n,
?(m(n),x) = m(n), m(n) = n+c,
(for all sufficiently large values of n), v/here c is an
integer independent of n and a (x) are independent K, n
of k, say, a (x). If the a (x) possess the property
that at a fixed point x £ [o,b]
v/here T(n) has the properties: I) T(n) 4= 0; 2)
lim T(n) = °o; 3) x(n) = o(n), n -* 00 , then we have n - 00 , .
for f £ i4^^+(x)
(7) L„(f,., - f(x) = ^ ^ ^ i ^ ^ . o ( - ^ ) .
Before .ob ta in ing convergence the. . rems fo r c l a s s e s
(2)
larger than the clasoes K„ „x(x) and lu T)+{^) of
functions, we note that no particular purpose is served
by taking n to be an integer in the definition of the
operators L . We assume, therefore, that n £ U where
U is an unbounded set of positive real numbers. \/ith
this stipulation the numbers m(n) also need not be
integers. Further, it v/ould be sufficient to assume
that the properties i)-iii) hold for all suffiently
large values of n.
The follo\/ing theorem extends the result of
tneorem 1 of Scnurer to the classes H p+(x) and
H„ P+<a,b>, witiKiut any extra assumption. THE OREL: 1 Let f £ K ;+(x), x £ [o,b]. Then for the
35
operators L defined in (l), there holds L^(f;x) -* f(x)
as n — oo ,
Also, if a, (x) - 0 uniformly in x £ [a,c]
(O g a < c g b) and if in (iii)p, (2) and (3) hold
uniformly in x £ [a,c], then L (f;x), (n £ U), converges
uniformly on [a,c] to f(x) as n -« 00 for each
^ e \,R+<"'^>-
Proof. For k g m, where k and m are positive integers,
we have
. m-i m-1 a
, m , , m k = k! s J-. -T-yr
1 = 0 (k-i"+i)'
where Q ~ , i=0,1,,..,m-1 are the Stirling's numbers
of second kind ([22], § 58, p. I68-I73). Hence for an
arbitrary positive integer m \/e have for k g m the
relation
(k)/ V k „ / ., \k (k)/ X k m-i r.^ . . ^ k ! L ™ L k" m-1 G ^ O ^ ( n ) x a„ (8) (-1) — Y i ;r—=.^^ (k-m+i)
X (n) 1 = 0 ^ , \ "" / -\k-m (k-m)/ \ k-m
m__m „ ^ (-1) m (x) X a X m-1 . ^ im m ? (n). -_nMm-(n),x)(l+a^_ i )(x))} ^ n) 1=0 » \ / X (n)
m-1 m-1 2
+ ^S 'n Mm^(n),x)(l+a, . u ^(x))} . x"'(n) i=0 k-i,mi(n)^ ^ I
/ .\k-m+1 (k-m+l)/ N k-m+1 (-1 ) m . (x)x ^ ^ ^ m-1/ N^ '
m (n) • (k-m+1)! + ... ,
v/here m (n), 1 = 0,1,2,... are defined inductively by
o m
(n)=n, m (n)=m(m (n)), m (n)=m(m (n)), and so on.
In the following we will use the easily verifiable fact
that a = 1 and a = m(m-l)/2. Omitted terms correspond m m ^ ' ^
36
to the values of i < m-1.
For the values of k g m-1, the only important
case is when k=m-1. In tnis case there holds
m-1 o" - (m-i)'" = (m-1)! E jf-jy ,
1 = 1 ^ ^'
and as in (8) we have
/ .Nm-1 (m-l)/ \ m-1
^^^ (^iryi m. ^ ^ X (n)
m-1 m-1 _ (-1)% (°) (x) x° a X m - 2 . ^ ' Tiii_-i\ / m n {?(m^(n),x)(l+a . . ±. Jx))] ^
._„' ^ \ /» /\ m-1-i,m-^(nj^ "' 0! x"^(n) i=0
"T* • • • •
Using" this and (S) we have
(10) 2 ( - 1 ) '
( k ) / N k -^ ( x j X ,,m
i n k ! ^f \ k=0 X (n )
m m a X oo m - i = — 2 [ n W ( m ^ ( n ) , x ) , x ) ( l + ( x , . ±f s ( x ) ) i ,
m, s , n . r^ \ /» /» / \ m + k - i , m ^ ( n ) ^ ^ ' ' X (n) k=0 1=0 ' ^ ^ / . \k ( k ) / N k ( -1 ) 9^^ ( x ) x ^m-1 ^m-1 ^ ^_^
. ^ ] + 2 [ n ^' x'^Cn) k=0 1 = 0
{ ? ( m ^ ( n ) , x ) ( l + a . , . ±f N ( X ) ) J , ' \ '» / \ m-1+k-i,m-^(nj ' ' *
/ . s k ( k ) / N k
^-'^ ^ m-V , ^^ k! ] + • • • !
where the omitted terms contribute a o(l/ x(n))
quantity,
From (2) and (3) it follows (since by (3) we have
37
m (n ) — oo as n -* oo) t h a t f o r each f i x e d 1 = 0 , 1 , 2 , . . .
(11) n — oo
, i , K 4 ^ = 1 . xTnT
From this and (iii) it follows that given an arbitrary
£ > 0 we can choose a positive number N such that for
all n > N we have
and
m-2 n
i=0 (13) 1-£ < ""ii j('^("^VM))(i+a , ^ . u (x))j < 1+£, ^ ' •-n x(^) m-1+k-i,m-'-(n)^ ''
We can choose N so large that if n > N then the
contribution of omitted terms in (lO) does not exceed
e/x(n) in absolute value. Then it follows from i), (10),
(12) and (l3) that if n > N
m-1 m-1 a X
(14) L (t ;x)-x < £x + j—T— (1+e) + —7—r . ^ ^ ' n^ • ^ I x(n) x(n)
Since e > 0 is arbitrary, we have
(15) lim L^(t"';x) = x'". n -* 00
Noting that m in (14) is an arbitrary positive
integer and that L (l;x) = 1 for all n, from theorem
1.1.2,2 we have
(16) lim L^(f;x) = f(x) , (x £ [o,b]) n -* 00
for each f £ H p+(x) where m is any positive integer
g 2. This is equivalent to the first assertion of
38
theorem 1.
If the conditions in the second assertion of
theorem 1 are satisfied then for x £ [a,c], I' in (14)
c m be chosen to be independent of x. It follows from
(14) that (15) then holds uniformly in x £ [a,cl and
the second assertion of theorem 1 follows from the
sec md assertion of theorem 1.1.2,2. This completes the
proof of theorem 1,
In the next two results we improve upon theorems
Il-lll of Schurer, Again we note that no extra
assumptions are involved,
THEOREM 2 Let the operators L defined in (1) be such
that v/e have the special case
x(n) = n+p, m(n) = n+c, il'(n,x) = n,
where for all sufficiently large values of n, p and c
are constants independent of n. Further if for all
sufficiently large n, a, (x) are independent of k, say
a (x), and satisfy
(17) a (x) = - e ^ + o(l) , n ^ 00 ,
(2) at a fixed point x £ [o,b], then for all f £ H); p+(x)
(18) L (f;x)-f(x) = 2(p(x)-p)xf'(x)+x(cx+l)f"(x).,^(l^ 2n
as n
Further, if (17) holds uniformly in x £ [a,c]
(0 g a < c g b), where p(x) is bounded on [a,c], then (2)
(18) holds uniformly in x £ [a,c] for each f £ Hl R+<S-,C^ Q, R
Proof. If ollows from (1) that for an arbitrary
positive integer m, we have
39
m m , o x m-1
L (t"';x) = - S n [(n+ic)!l+ ^^4^ + o(l)j] n^ ' ^ / sm . „ '- '' n+ic ^n''-'
(n+p) 1=0
m-1 m-1 a X m-2 m
(n+p)'" 1=0 n [(n+ic)ji + - a ^ + o(^)j]+o(-i) „ '- ^ ' n+ic °^n^ 1J ^n'
m m ^ fn+ic+p(x)+o(l), a X n ^—^ ^—'-] m . r. n+p '
1 = 0 m-1 m-1 „
a X m - 2 . /• \ ^ ^ \ y, S n ,n+ic+p(x)+o(l), ^^U
n+p ._„ ' n+p ' ^n^
= o™ x"" (1 + ^i^-^)'^ + "i(p(^)-p) m ' 2n n
m-1 m-1 m /1 \
n 'n'
Thus, putting the values of a and a , for ' ram'
m=0,1,2,...
(19) L^(t'";x) = x'" + " ("'-1)( +I)x"'~
+ I^(P(^)-P)^'" + o(l).
From (19) and the first part of corollary 1.1.3.1
the rel:ition (I8) is immediate.
To prove the second part of tiie rem 2, it is
easily verified tnat under the given uniformity
conditions (l9) holds uniformly in x £ [a,c]. Thus the
second part follows from (19) and the second assertion
of corollary 1.1.3.1.3-[1 ] [1 ]
Let RL jj!(x) and Ht i[a,b] respectively denote the Si«,A y,A Til
totality of functions of the classes HL J ( X ) and •m,X'
40
[1 ] H ^[a,b], m varying over all positive numbers. m, A
V/e have the following generalization of theorem IH
THEOREM 3. If in the statement of theorem 2 the condi
condition (l7) is replaced by
(20) a (x) = -44 + oi-r^) , ^ ' n^ ' x(n) ^T(n)' '
where T(n) 4= 0, lim T(n) = <», T(n) = o(n), n - oo, n - oo , ,
then \/e have for each f £ Hl v+^-^^
(21) L (f;x) - f(x) = P lf'l" + o{-K), n oo, R~^ -" ^ TUT'
Further, if (2) holds uniformly in x £ [a,c]
(O g a < c g b) where (x) is bounded on [a,G], then
the convergence in (21) is uniform in x £ [a,c1 for (2)
each f £ H^^+<a,c>,
Also, in tne first assertion above, the class
n\ /}+{x) can be replaced by the larger class H|- j+(x) U,n / s Q,K
and in the second asserti.,n the class R \ pj.<a,c>
can be replaced by the class H^ J^[a,c].
The proof of theorem 3 proceeds on the lines
similar to those of the proof of theorem 2. After
pr. ving the first two assertions, we apply remark
1.1.3.1 made in connection with the relation (1.1.3.21)
to obtain the third assertion.
1.1.5 C-eneralizations for functions of several variables
In this section we obtain generalizations of some
of the results of earlier secti ns, suitable for
studying the approximation of certain classes of real
or complex valued functions defined on a subset of a
41
Euclidean m-space R by means of a seauence of linear m
positive operators. Proofs of the result of this section
can be carried out on the ba,sis of the notions
intr-^duoed in earlier sections and along the lines of
the proofs of results in ([62], chapter 2) and [59]. Definitions, conventions -.nd notations
Let the ra-tuples (n.,n^,...,n ) and (p. ,p,.,... ,p ), 1' 2 ' ' m ' \x--|Fx-2» '-^m '
where m is a natural number, be denoted by the symbols
II and p respectively. This notation is consisitent with
that of Schurer ([62], chapter 2). 'v/e shall always
assume that II,p £ R" , the first hyperquadrant of R .
The notation R — oo signifies that n. — oo, j = 1,2,...,m.
Let U d R be such that the set! min n.: II £ U } "•"•'" 1 g j gm - "'
is unbounded. Let lL.,,N £ U \ denote a class of linear ' N' m'
operators mapping a linear space D(X ) of real or complex valued functions defined on the set X d R
m — ra into a linear space D(X ) of real or complex valued functions defined on the set X ^E_ R . V/e assume that
m m if f £ D(X ) then f, the complex conjugate -jf f, also belongs to L(X ). The symbol X in D(X ) represents an ° ^ m' •' m ^ m' - assumption that for f £ D(X ), II £ U and
^ ^ m ' m X = X(x-,x„,...,x ) £ X we can construct L (f;X)
\ -|» 2' ' m' n^ ' ' provided the values f(H), 3 = zi^^ ,1^, . ..l^) £ X^
.are given. An operator L,., II £ U , is said to be _ N' m'
positive on a set JT cz. X if for each f £ D(X ) the m — m' ^ m^
assumption f(H) g 0 for each E £X leads to L,.(f;X) > 0 ^^' m 1\^ ' ' ~
for each X £ X . The class (L,-, K £ U j is said to be m ' ir ^ m'
ultimately positive on a set X S: X , if to each m m '
f £ D(X ) and satisfying f(E) s 0 for each S £ X , m ./ o \ / m there exists a natural ntunber n such that L (f;X) g 0
for each X £ jt whenever min n. > n. The set 7. 1 g j g m J
42
may consist of a single point X £ X or may contain
more than one point.
In the sequel we shall deal with the following
classes of functions:
H : the class of all real or complex valued functions P,A f(H) defined on X for each of which there exist ^ ' m
positive constants C,D, say, such that there holds
m p. f ( H ) g C + D 2 |^.|^, for all S £ X .
j = 1 '^ ^
K „ (H): subclass of H ,. consisting of the functions P , A P , A ' m • ' m
f(s) which, with an extension, are continuous at the
point 3 = X. (2)
H^ / (x): subclass of H consisting of the functions p.A P«X - ' m • ' m
f(3) which, with an extension, are twice differentiable
at the point 3 = X, in the sense tliat at the point X
all the first and second partial derivatives of an
extension of f exist and there holds m
f(3) - f(X) = 2 (l.-x.)f' (X) j-1 ^ ^ ""j
., m m p + j 2 (^-x )(^ -X )f" (x)+o( 2 ( .-x ) )
m p if 2 (E.-X.) tends to zero.
j = 1 ' '
Let S c=-E , Denote by H „ (S ) the class of all m — m '' p,X ^ m'
• ' m functions f £ H ^ which with an extension are
p,X " (2) continuous at all points X £ S . By H^ /. <S > we denote m p,X ra ^' m
the class of all functions f £ H ^ which with an p,X ^ ' m extension are unif-^rmly twice d i f f e r e n t i a b l e on S , i . e .
m' *
43
twice differentiable at all points X £ S such that m
the above twice differentiability relation holds
uniformly in X £ S . fl 1 ^
R^ 4 (X) : subclass of H ^ of functions f(H) which P tA PfA • ' m ' m with an extension possess all first order partial
derivatives at the point X and in addition satisfy
m m p f(H) - f(x) = 2 (^-x.)f' (X) + 0( 2 ( .-x ) ) ^ a = i ^ ^ ""o j = 1 ^ ^
if 2 (t.-x.) tends to zero. The class of all
functions f(3) £ H"- ^ (X) for each X £ S and for ^ ' p,X ^ ' m
' m which this relation holds uniformly in X £ S is
ril " m denoted by E^ 4 [S ].
p.A ^ m-' •^' m
If for some p, f £ H (X) we say that P , A
f £ E^^ (X). The classes E^^^ (x), H^^^ (x), ' m ' m ' m
H„ ^ (S ), E),^ <S > and H^''^ [s ] are defined in the Q,A m fi.A m Q,X '- m-' ' m ' m ' m
same way,
In the direction of approximation of functions of
many variables Schurer [62] (also see [59]) gave the
following two theorems.
THEOREM I Let H ( X ) denote the class of all real
functions f(x) which are defined in R and which have ^ ' m
the properties
1) f(X) is continuous for X=X ,
2) f(x) = 0( 2 X.) when |x.| - oo (j = 1 ,2,... ,m).
j = 1 ^ ^
Let f(x) £ H(X ) and let jL j (n=1,2,...) be a sequence
of linear and positive operators defined on H(X ). If
we write
44
L^(1;X) = 1 + a^(x),
L (t.;X) = x + p (X), (j=1 m; n=1,2,...) n J J _ 'i»J
Ul p 1" p
L ( 2 t;;X) = 2 x; + Y„(X), j = 1 '^ j = 1 "
and if a (X), p .(x) and Y (l ) have the property that n '' n, J ' n^ ^ ^ •'
lim a (X ) = lim P . (X ) = lim Y (X ) = 0 n^ o' n,j o' 'n^ o'
n-*oo n-'oo '" n->oo (j=1,...,m), then v/e have
lim L (f;X) = f(X ). n ' ^ o' n -* oo
TiIEOREM II Let H^^^(X ) denote the class of all real
functions f(X) £ H(X ) of which all the second
° (2)
derivatives exist at the point X . Let f(x) £ H^ (X )
and let {IT J te a sequence of linear and positive
operators defined on H(X ). If the operators L,. have
the pr.iperty that in a fixed point X L„(1;X ) = 1 + o( / 0 iN^'o' ^cp(n.)'
Y. -(X ) + '^ X + o(^^)
9(n.) >(n.)' L„(t.;X ) = X .
(k,j=1,...,m; k + j)
L(t2;x ) =x2. , ! 2 i i ^ ^ o ( - ^ ) N' j' o' oj 9(n.) >(n.)'
J 0
L,,(t, t. ;X ) = X X . ir k j' o' °k ''
-(XJ
^2-k i^^o) 9(n^)
2:i,k^ o' f 1 N t'\^^_^\ + y \ + o(—7 v) + o(~7 V) »
9(n^) >(nj)^ >(j^k)
where 9(n.) + 0 and 9(n.) -* oo when n. — oo (j = 1,...,m). J J J
If there exist positive integers p. (j=1,...,m) such J
that
45
then we have
[?, .f +4('i'o • •-2X .?, .)f" ] m L i;j X. 2^ 2;j,j oj 1;j^ x.x.J
L,,(f;X J-f(Xj= 2 ^ ^ ' ^-L_ N^ ' 0 ' ^ o' ^ 9(n.
m 'I'o.v .-x„, ..v ' 2;j,k~ o 1;j •^i 2 p;k,.1 o.i-1;k, '-> k '- ., ^,,(1 )
0 +k
where the values of all functions ¥, f and f" are
taken in X , o
Remark Theorem II of Schurer is not correct. It
requires a modification. In fact a mere assumption on
the existence of all the second derivatives at the
point X is not enough. Following is a counter example.
In Rp, let an operator sequence JL j, (n=1,2,...),
be defined as follows:
Vf;(x,y)) =lf(x+J, y+^) -Jf(x-f. y-J).
n=1,2,..., where a and b are two positive numbers,
We have
\{M{x,y)) = 1,
L (t;(x,y)) = X,
and
\is',ix,y)) = y, \ { i ;(x,y)) = X + (a/n) ,
L (s ;(x,y)) = y + (b/n) , 2 L^(ts;(x,y)) = xy + ab/n ,
V(t-x)2^-^2.(x,y)) = (a/n)2'"+^
L^((s-y)2^^S(x,y)) = (b/n)2^+2^
46
for an arbitrary positive integer m.
The asymptotic formula given by theorem II becomes
L^(f;(x,y)) - f(x,y)
= iKg)^f;;,(x,y).(^)2f;,(x,y)j
Consider the follov/ing function
0 , (x,y) = (0,0)
f(x,y) = \ (. 2 2 xy ^ 2^ 2 » otherwise,
X +y
defined on the xy-plane. At the point (0,0) all the (2)
second derivatives of f exist and f £ H^ ((0,0)). We
have
f(o,o) = f^(o,o) = f^(o,o) = f;^(o,o) = f^y(0,0)=0,
f;;y(0,0) = -1 and f^^(0,0) = 1,
If theorem II would be correct we should have then
L^(f;("0,0)) = o ( ^ ) . n
However, an actual calculation shows that
n (a +b )
For an arbitrary choice of a and b the two results
are clearly incompatible,
As we shall see in the sequel, a correction, to
render theorem II applicable, would be to assume
further that f is twice differentiable at the point X ,
In the above counter example the function f is clearly
not twice differentiable at the point (0,0).
47
The following thejrem is a generalization of
theorem 1.1.2.1, a theorem of Volkov [77] and theorem I
of Schurer. It gives the basic approximation result
for the classes H „ (x) and H ^ (S ). p,X p,X m ' ra ' m
THEOREM 1 Let p and p'(p! < p., j=1,2,...,m) be two J J
m-tuples of even and odd positive integers, respectively. Let{L„,N £ U j be a class of linear operators defined
' N' m' on a common domain D(X ) (X <- R ) of functions into a
^ m m — m domain D(X )(X c;- R ) of functions and ultimately
^ m' m — m' ^ -ni r '* P-;
positive on a set X i^X. Assuming that 1, E,.>J m p. J
(j = 1,2,... ,m) and 2 •'' £ D(X ) and writing j = 1 ^ "
L,,(1;X) = 1 + a.j(x)
p! p! (1) h {l/^',X) = x ^ + p (X), (j = 1,2,...,m), and
N Jj a i.,Pj m Pi ^ V-
L ( 2 ^ • ;X) = 2 X J + y (X), n j^., J j^., J iJ,P
where X £ X and N £ U , i f and only i f there hold m m
(2 ) l im a (X) = l im p (X) = l im Y:- ^ ( X ) = 0 , N - oo ' N - o= ^"'Pj N - c-- ^"'P
j=1,2,...,m, then for each f £ D(X ) , K „ (x) we have ^ ' ' ' ' ^ ra' p,X ^ '
'• ' m
(3) lira L (f;X) = f(z), N - oo
Further, let S ^ X be a compact set. Then for ' m m ^ each f £ D(X ) r H (S ) relation (3) holds uniformly
m p • A m ' m
in X £ S if and only if (2) hold uniformly in X £ S .
Remarks 1-3 made in connection with theorem 1.1.2.1
have the follov/ing analogues; m p.
Remark 1 Let 1, 2 (E,.-X.) ^ £ D(Z ). Then in tiie
48
statement of theorem 1 relations (2) can be replaced
by the following
m p. (4) lim {L (l;X)-1j = lim L ( 2 ( .-x.) ;X)=0.
N - o o " ' N - o o ' ' j = l J > J
Remark 2 In the case of functions of many variables
there is the following generalization of Hsu's theorem
[17] for a single variable.
THEOREM 2. Let la j be a set of positive real numbers ' n . '
J increasing to infinity with n., j=1,2,...,m, N £ U . Let II.., N £ U I be a class of linear operators
' N' m^ L». : D(X ) -» L(X ) and ultimately positive on X where N ^ ^ m' ^ m' "" m
X ,X S R and X •^ X . Let X contain an open m m m m m m
neighbourhood of the point zero in R . Let the p'. p. m
functions 1, ^.J, ^.0 £ D(X ), j=1,2,...,m, where p and p'(p'. < p., j = 1,2,...,m) are two m-tuples of even
J J and odd positive integers, respectively. Further let the limit relations
k. _ k. (5) lim L ((a ^ ) ^; aJ x ) = x ^, j=1,2,...,m,
N - 00 '' "j J "j J J
where k.=0, p!, p. (j=1,2,...,m), exist and hold J J J
uniformly for all values of X in every bounded sphere
of R . Let f be a real or complex valued ftuiction m
defined on R such tha t for each II £ U the function n m ^^V^l' V 2 "n^m)^'^(\)-
1 2 m Then, if f £ H „ (x) where X £ R , there holds
' p,X ^ ' m'
the limit relation
— 1 — 1 (6) lim L„(f(a ^.,...,a ^ );(a~ x.,...,a~ x )) ^ ' ,, N^ ^ n. 1' ' n ^m" n^ 1' ' n m^'
N — CO 1 m 1 m
f(x).
49
Also, if S is any corapact subset of R and
f £ H V (S ), then (6) holds uniformly in X £ S . p,X^ m m
Remark J In the special case when X c:- R , we have
the follov/ing generalization of theorem 1.
THEOREM 3 Let X , X ^^ R" and let p and p' be two ^ m' ra m
m-tuples such that 0 < p! < p., j=1,2,...,n. Let J J
fL.-,II £ U j be a class of linear operators ' H' m' ^ L, : D(X ) -* L ( X ) and ultimately positive on a set IN m m -n I
X SE. X . Let the functions 1, .J (j = 1,2,...,m) and m ra f •'j \o » » I /
m p. 2 .' £ L(X ). V/riting
0 = 1 J ' " '
L^j(l;X) = 1 + aj.(x),
p! p' . (7) Lj .( J;X) = x^ ^ + Pj^^p,(x), j = 1,2 m,
J m p. m p.
L ( 2 ?; J;X) = 2 x ^ + Y: p(X) j = 1 ^ j=1 ^ ^ 'P
where X £ X and II £ U , i f and . nly i f m m ' ''
(8) lira a (X) = lim p (x) = lim y „(x)=0, N - oo •"' II - oo - ' ' P j II - oo ">^
j = 1 , 2 , . . . , m , t h e n f o r each f £ PI .. (X) r D(X ) ' ' ' ' p , A ^ ' ^ m ' i_ ' m
we have
(9) lira L ( f ;X) = f ( x ) . II - oo '
F u r t h e r , l e t S S^ X be a c^mpo-ct s e t . Then f o r ' m m ^ each f £ Il .. (S ) / I)(X ) r e l a t i o n (9 ) nolds u n i i ^ r m l y p,A m" ^ m' ^ ' •' i n X £ S i f and only i f (8) ho ld u n i ; j r n l y i n X £ £ . m m
Theorem 4 be lo \ / i s a j ^ e n e r a l i z a t i o n of ti ieorem
1 . 1 . 5 . 1 . I t a l s o g e n e r a l i z e s and improves ti:iejrera I I
of L c h u r e r ,
50
THEORni 4 let p. > 2, j = 1,2,...,m and let {L.„iJ £ U j
be a class of linear operators L,. : D(X ) -- D(X ) and L ^ m' ^ m'
ultimately positive on X S^ X where X ,X ^. R . Assume m m m' m m
that the functions 1, •, • , (j,k=1,2,...,m) and m p. ^ ^ ^ 2 U--X.1 "^ (where X varies over X ) £ D(X ), With . I ' J J ' ^ m' ^ m' X £ X if md only if there hold the relations
m
+ S o ( — 7 — r ) , L (1 X) = 1 + D J r- + s o( 7 ! N • - i 9 - ( n . ) . . ^m. in. ,
1 = V 1 ^ l ' 1 = 1 ^ 1 ^ 1 '
m 7 . . (X) m
(10) m <!' , , ( X )
L„(^^;X) = x^ + 2 ^'^^ V + 2 o(—^ r"
L , ( , , , . ; X ) = x , x . + ^ 2 ^ ^ ^ ^ ^ E ^ o ( ^ ) ,
(k j ) ,
L ( 2 \i.-x \ ^;x) = s o(-Tr-y),
j , k = 1 , 2 , , . . , r a , a n d m p .
( 1 1 ) L,,( V Ir _^ I J
J
a s II — CO , w h e r e 9 . ( n . ) 4= 0 , cp. ( n . ) ^ 00 v/hen n.-* 00 ,
i = 1 , 2 , . . . , m , t h e n f o r e a c h i £ H ( 2 ) ( X ) ^ D(X ) t h e r e p, A m •^' m
holds the asymptotic formula m
(12) L.-(f;X) - f(X) = 2 ^ - ^ [^,i(x)f(X) 1=1 ^ 1 ^ 1'
+ 2 {(V,..(X) - x.Y .(X))f' (X) + ^(V..-(X) . H 1ji J 01^ '^ x.^ ' 2''''2.ii ' 0 = 1
^j'"1j: - 2x.'i', , . (X) + x^? .(X))f" (X)l
J Iji J 01 ^ x.x.^ ^ ' J J
51
+ ^ 2 ( Y^, ..(.X) - X, T, ..(X) - x.'?-, . (X) 2 _ ^ 2kji^ ^ k Iji^ ' J Iki^ '
k, 0 -1 k + j
m + X, X. ^ .(x))f" (X)] + 2 o ( — r),
k J 01^ ^ Xj x ^ J ^^^ ^9^(n^)^'
as N -* °o . Further, let S be a bounded subset of X ' ra m m and assume that t e function '?(X) = 2 {?p..(x)
i,j=1 ^^^
- 2x.Y,..(X) + x^T .(X)j is bounded on S , Then, in J Iji^ ^ 2 o±^ m '
order that (12) holds uniformly in X £ S for each (2) ™
f e H^ v^S > r\ D ( X ) , it is necessary and sufficient p.X m ^ m'' •' ^' m
that (1) and (II) h^Id uniformly in X £ S .
Remark 1 If p' = (p', Pp,..., p') is an m-tuple such
that 2 < p'. < p., j = 1,2,...,m, then in theorem 4, J J
relation (II) can be replaced by the follov/ing
m p! P • -P '• m (13) L .( 2 U-x M ( U j M .1);X) = _2 o ( ^ - ^ ) ,
J = 1 ^ >J >J 1 = 1 T^V -[_/
as N ^ 00 , Here we have asstimed that the function m p! P • -P '• 2 k.-x. I ( U. I "^ -^+1) £ L(X ) for the points X . ., ' J J ' ^ ' J ' ^ m^
under consideration,
COROLLARY 1 Let p. > 2, (j=1,2,...,m), be even J
positive integers. Let {L,., N £ U 1 be a class of ^ ° ' L' _ m' linear operators L, : D ( X ) — L ( X ) and ultimately
~ J^ ^ m' _ m' ^ A positive on X cz X where X ,X ^ R . Let 1,^.,E,.,^.,
m m m' m m j J J P-:-4 p.-3 p.-2 p.-l p
^1,^^,^^^ ,^-'^ fl^'^ fl-'' and l . \ (j,k=1,2,...,m),
£ L(X ) , If X £ X , then in order that (12) holds for m' /pN m'
each f £ H^ .;> (X) r\ D(X ) , if is necessary and p.X ^ ' ^ m " " •^' m
sufficient tnat it holds for the above mentioned
52
functions, Fti' thp'", i'" S i s J, b-undsl ub "t of X ' m -1
and the function
ra P ¥(X) = 2 {l' ..(X)-2x.1'. ..(X) + X ¥„. (X)!, ^ ' • --1 2ji^ ' J Iji^ ' J 01^ )»
1 , J - I
where I',,..,*!'... and ? . are as in (l2), is bounded on 2ji' Iji 01 ^ "
S , then in order that (l2) holds uniforraly in X £ S m' (2) "" for each f £ H^ .;( <S > r L(X ), it is necessary and p, X m ^ m " "'
•^' m
sufficient that it holds so for the functions mentioned
above.
COROLLARY 2 Let p . > 3 , ( j = 1 , 2 , . . . , m ) , be p o s i t i v e J /\
i n t e g e r s and l e t X , X ^ R+. Let JL^, N £ U jbe a c l a s s ^ m' m m '^N' m'
of l i n e a r o p e r a t o r s L, : D(A ) -• D(X ) and u l t i m a t e l y
p o s i t i v e on X ^ ^ X ^ ^ . Let 1, ^j , ^ ^ ^ ] , ^k^j . ^ j ^ . ^ j ^ . p.-2 p.-l p. ^^J , -i,^ and l ^ , (j,k=1,2,...,m), £ J(X^), If
X £ X , then in order that (l2) holds for each m' ^ ' (2)
f £Ii^ ' (x) /-^D(X ), it is necessary and suffient that p,X ^ ' ^ m'' " •^' m
it holds for the functions enumerated above. Further,
if S is a bounded subset of X and the function "(X), m ra \ / f
as defined previously, is bounded on S , then in order ' ra
that (12) holds uniformly in X £ S for each (2) "^
f £ H^ . <S > r> D(X ), it is necessary and sufficient p,A m ^ m " •' ^' m
that it holds so for the above enumerated functions,
Remark 2. Referring to theorem 4, let us consider the
special case when there holds the relation
m P
(14) V(X) = _ 2 1^2ji^^)-2''/iai^^^ + /oi^^M =0-
1, J = 1
In case the relations (IO) are satisfied, as a
consequence of (14), ne can easily show that
53
(15) ?. . . . ( X ) - X, V, . . (X) - x . T , , . (X) + X, X. V . (X)=0, ^ -^' ^KJl^ ' k I j l ^ ' J I k l ^ ' K J 01^ ' '
k + 0 , i , j , k = 1 , 2 , . . . , m . Thus t h e formula ( l 2 )
r e d u c e s to
m (16) Lj^j(f;X) - f(X) = 2 — r ^ [ ? ^ . ( X ) f ( X )
i=1 ^1^ ±' i l l H i ^
+ 2 ('?, . . ( X ) - x . ¥ . ( X ) ) f ' (X)] + 2 o ( — v) ^ I j i ' ^ J 01^ ^^ x ^ ^^^ > i (n j_ ) ' ' j=1 •" " " "J
il -* oo, in which the second order partial derivatives
of f do not occur,
Analogously, as in the case of a single variable
(remark 1 following corollary 1,1.3.1.3), here also
it is not necessary to assume the existence of the
second partial derivatives and it is sufficient to
[l 1 ass-ome that f £ H"- j (x) r\ D(X ). Also, in the case of
P'\ '^ uniform convergence of (l6) for X £ S , it is sufficient
Til "* to assume that f £ K'- ^ [S ] r D(X ),
p,X '- ra-" ^ m' ' m
Further, we note that if (14) and the first three relations of (l0) hold, then we have
m p m (17) Ljj( 2 (^-x ) ;X) = 2 o ( ^ - ^ ^ ) ,
j=1 "^ '^ 1 = 1 ^1^ 1^
and irrespective of whether (ll) holds or not, it can
be shown that (l6) holds for each f £ H'-]-' (x) r. D(X ), ' P ,X ^ ' ^ m^'
• ' m where p' = (2,2,,..,2). Also when (14) holds for each X £ S and the first three relations of (10) hold m ^ ' uniformly in X £ S , it can be shown that (I6) holds
^ Til uniformly in X £ S for each f £ HL ,J [S ] L(X ).
m p.X ^ m-* ^ m^ m
Lastly, it is to be noted that v/hen (14) holds,
in order to prove (16) it is not necessary to assume
that \ \ ^ t (k + j;j,k=1,2,...,m), £ L(X ) and the
54
fourth relation in (lO) can be discarded altogether.
THEORSI-1 5 Let p. g 2, (j = 1 , 2,... , m ) , be even positive J
integers. Let JL,., N £ U j be a class of linear ' ' IM' m^
operators L„ : D ( X ) -» D ( X ) and ultimately positive ^ N ^ m' ^ m' '' ^ on X where X , X d R and X ci. X . Assuming that
m* m — m m — m 2 Pi~2 Pj-I Pj , N / N
^»^y^y ^j . ^j and ^^^ (j = 1,2,...,), e D^xj and X £ X , if and only if there hold
m' k. k. m
(18) L„(^-'';X) =x.^ + 2 0(— ] v). 1 1 - ^ , ^ ^ N^ J ^ J i ., 9i(n^)^'
j=1,2,...,m; k. = 0,1,2,p.-2, p.-l, p., then for each r-t 1 o o J J
f £ HL'-! (X) ^ D(X ) p,X ^ ' ^ m' ' m
m (19) L (f;X) - f(x) = 2 0( 0 , N ^ - .
i=1 ^i^ ±^
Further,if S is a bounded subset of X , then (l9)
holds uniformly in X £ S for each f £ H'-''j [S 1 r^D(X„), "' m p.X I- m-J ^ m"
if and only if (18) hold uniformly in X £ S . m " +
Also, when X , X c^R , the above results remain ' m' m —• m'
valid when p. g 2, (j=1,2,...,m), are positive numbers, J
not necessarily even positive integers.
1.1,6. On a generalized sequence of linear positive
operators.
Analogous to the sequence of the operators L
defined in (1.1.4.1) for functions of a single variable,
Schurer [62] considered operators L„, for real valued
functions of several varaable, defined as follov/s: i-+...+i^ - i. i^
/ -\ 1 m I/v^ 1 ^ oo oo CO (-1) 9 | A X ) X ...X
(1) L (f;X) = 2 2 ... 2 . , . , ., , ^^ i,=0 i^ = 0 i =0 ^r ^2 ^m'
1 2 m
55
„/ 1 2 ra ^
•^^7;T^'^?V^ 71:^^' where for each function 9|,T(X) the follovvring properties
are satisfied in a domain K of the first hyperquadrant
of R . m 1) 9|^(X) can be expanded in a Taylor's series in a
closed region K , K is the union (when X runs through
K ) of the closed spheres with centre at X and radius |x|; 2) 9i (o) = 1;
i +...+i -, 5) (-1) 9i(x) S 0, (i 1 =0,1,2,..,;X £ K);
4) there exist positive integers n. (j=1,2,,..,m) not J
depending on i ,i2,,..,i , such that in the region K we have
I-E. (2) -9d(X) = ?.(n.;X)9,- ,'^ —>,„ (X), ^ ' ^N^ ' J J '^lM-(n.-n. )E. '
J J J
• ^'' " " i n ^^^^ ' ( j = 1 , , , . , m ) ,
where a) a. (x) ( j=1 , , . , , r a ) converges uniformly
in i . to zero in K when n. — 00 ; J J
b) there ex i s t the pos i t i ve functions x- (n- ) J J
( j = 1 , . . . , m ) (used in (1)) (monotonically) increas ing to i n f i n i t y as n. — 00 with the property tha t in K
J ¥ (n ;X) ? . ( ^ ; X )
(3) lim ' z* V = lim - V ^ S — = 1 , X•(n . ) V . (n . J
J J
J i ^ + i 2 + . . . + i ^
By 9jj we mean , , 9^ ' 1 2 , III
o x , 0X_ . . . o X 1 ^ m
56
I = (i-i ij^). 1-Ej = (i-,,...,ij-1,...,ijj^), and
N-(n^-ir)E^ = (n^,..., n^,...,nj^).
In above we shall, however, allov/ n., n., (j = 1,.» ,m) J J
to assume arbitrary positive values not necessarily
integers. Moreover, as we shall be concerned with an
asymptotic behaviour of {I'^T] it would be sufficient
to assume that the properties l)-4) are satisfied when
min n. is sufficiently large, 1 g j g m ^
Similar to theorem 1.1.4.1, in this case we have
the following basic convergence result, which
generalizes and improves the result of (Theorem 10,
Schurer [62]) under the same assiunptions.
THEOREM 1. If f £ H^ R+^^) "* ^ ^^^ functions f^{X) * m
satisfy the above conditions l)-4) then L^(f;X) defined in (1) converge to f(x) when N - oo and X £ K. If,
moreover, a. (X) (j=1,...,m) converges to zero
uniformly in X £ K' K where K' is a bounded and
closed domain, and if the relations in 4) h) hold
uniformly in the same domain then L (f;X) converge
uniformly in X £ K' to f(x) for each f £ H„ T,+(K'). ' m
The following tv/o theorems improve the result of
(Theorem 14» Schurer [62]). They also generalize
theorems 1,1.4.2 and 1.1.4.3.
THEOREI'I 2. If, for min n. sufficiently large, the 1 g j g m '^
operators L„ defined in (I) have the special properties
that
X.(n . ) = n + p , (p being a cons tan t ) , J u J J J
(4) n. = n. + c. , ( c . being a consteint), J <J J *}
57
'?(n ;X) = n , J J
P,(X) . (4) a._^_(X) = a^ (X) = ^ — + o ( - ) , n ^ e c ,
0 J 0 J J
(j=1,2,...,m), where p. are functions in A, then for ( 2)
each f £ 'A\ 1+ (X) and X £ K we have 0 , J
m m
(5) L„(f;X) - f(x) = Z — ((p.-p.)x.f' J ' J J
1 2 ' 1 + 77 (x.+c.x.)f" j + 2 o(-i-),
2 ' J J j' X X 1 ^ n ^ '
as N -• ooj v/here the values of all the functions occurin
in (5) are taken at the point X.
If, further, (4) hold uniformly in X on a bounded
domain K ' S K with p.(x) bounded on K', then for each (2) s J
f £ H^^+<K'>, (5) holds uniformly in X £ K'. ' m
THEOREM 3» If in place of the last condition in (4) we have
P.(X) ., (6) an.(x) = - V ^ + o(—/—r) ,
'^ T^(n^) ^T^(n^)^ '
where T.(n.) =1= 0, T.(n.) -'oo, T.(n.) = o(n.), n. -«=
(j=1,...,m) then v/ith the remaining conditions of (4)
intact, there holds for each f £ E'- J+(x), X £ K, the -, , . ' m
relation x.p.(x)f' (X)
m j^j^ ^ n.^ ' m
(7) L ,-(f;x) - f(x) = z ^ ( y + .\-(Tih J=1 G J J=1 r J
a s II -> 00 ,
Further, if p.(x) (j=1,2,...,m), is bounded on a
boujided domain K' K and (6), (j = 1, 2,... ,m), holds
uniformly in X £ K', then (7) holds uniformly in X £ K'
for each f £ RL''j^. K' ,
58
Applying these theorems for the three types of
operators given below: i . - n . X . i .
oo oo m n .^ e "^ ^x."^ (8 ) L ( f ; X ) = 2 , . . 2 H -^ r—, ^ ,
i^=o i^=o j= i y i-, i
f (— —) ^n- ' ' n 1 ra
oo oo m n . ( n . + l ) . . , ( n . + i . - 1 ) (9) L ( f ; X ) = 2 . . . 2 n - J - ^ ^ . , ' ' ' ' .
i^=o i ^ = o j = i -y
- ( n . + i . ) i . i . i . (1+x ) J J x^ f ( - l - f ) , and
" "J 1 m
oo oo (n+a)(n+a+1 ) . . . ( n + a + i + . . . + i -l) (10) L (f,.X) = 2 , , . 2 T-, T -7 - J S —
^ i , = o i =0 ^^• • •* "-m-1 m
f, ^ -(n+a+i^+,.,+i^) i i i i ,(l+x^+,.,+x^) x ..,x^ ^ ( T ' - ' — ' '
(a is a positive integer or zero), which are defined
for X. g 0, (j=1,,..,m), we have, respectively, the
relations
x.f" (X) m J X ,x. ' m
(11) Lj^(f;X) - f(x) = 2 ^ + 2 o(^),N ->.x>, j=1 j j=1 j
x.(l+x.)f" (X) j^ 3^ x.x "• '
u u 2n. • „ "•
J=1 0 J=1 J
m y 0' i^i ^ 1 (12) L^(f;X)-f(X)= 2 2n. +.^ °(^)»^ ~'
and
1 " (15) L„(f;X)-f(x) = - [ 2 X. {2af' (x) + f" (x)i
J=1 J J J m
• . J , j^k x.x^^^)] +°(i)' ^-*-' J,k=1 J k
valid for f £ YS^IPC), x. g 0, (j = 1,...,m), and that ' "m
59
they hold uniformly in X on any bounded region K'^ R" (2) ^
for each f £ HJJ^^+<K'>. ' ^ (2)
Remark 1. For the classes H^ (X^) (as defined in
theorem I.I.5.II) the formulae (11)-(13) were obtained
by Schurer [62]. For a correct interpretation we refer
back to the remark made at the end of theorem 1.1.5.II.
Remark 2. Let us note that in the case of operators
(10) a direct application of theorem 2 is not possible.
It is due to the fact that in this case condition 4)
is not satisfied, as not only there is a change in the
j-th component n. but that all the components suffer a J
change. The essential effect of it is an addition of the quantity x,x./n to the function f^v • °^ theorem
1.1.5*4 and consequently the quantity
. m x, X. 1 k i 77 2 "• f" must be added on the right hand side 2 , . . n X, X. ^ k,j=1 k J
k + j
of (5) in theorem 2 to obtain the correct expression,
ignoring the non-satisfaction of 4). Thus we arrive at
(13).
Remark 3. In fact in the same way as in remark 2 above,
applying theorem 1.1.5.4 we can prove results similar
to those of theorems 2-3 for the operators in (1) for
which more than one component of N may suffer a change
in 4)» Thus instead of (2), if we have
I—E (14) -9j(X) = j(n.,X)9 I ^ (x).(l+ai_^_(X)),
k=1 j
where the functions ?.(n.,X) and a. (x) satisfy the J J
conditions a) and b) as before, then if for all N, with
60
min n. sufficiently large, we have n, =m,+C , 1 = j = m ' j j
where C are fixed constants (j,k=1,...,m), with all j
other evaluation of L„(t.t, ;X) shows that
C. C. (15) ^ ^k ,. , , V ^ ' ~r "^ ~r ' (J.k=1,...,m),
Thus the components suffering a change are related
(and in particular if all the components do change, we
can take n. = n.(n), j=1,...,m, and the operators
L„ can be thought of as some operators L of a single
index). Now, the required asymptotic formula will be
obtained by increasing the right hand side of (5) by the
quantity C.
1 " ^k •k 2 — - X, X. f" 2 , . . n. k J X, X. k,j = 1 J ' k J k 4= j
and replacing C. in (5) by C. (since the role of C. is
now taken over by the constant C. , j=1,,..,m),
The result of theorem 3, however, remains valid,
without any change, for the operators (1) with (14)
instead of (2) and n, = n, + C, , (j ,k=1,... ,m), j j
and the other conditions remaining the same.
1,1.7 A method of constructing operators for functions
of several variables.
Theorems 1.1.5.1-1.1.5.4 not only enable us to
study convergence properties of a knov/n sequence of
linear positive operators but they also provide us with
a method of constructing such operators for functions
of several variables with the help of those for ftmctions
61
of a single or fewer variables.
Thus, for instance, let {L^j,{L^ },..., |L^j be m
sequences of linear positive operators each defined
for functions of a single variable. Renaming these by
L (-;x ), L (-;x ),...,L (-;x ), consider the "l 2 m operator L^(-;X) constructed as follows: Let
f = f(H), 3 = ( ,^2 ^m)* Operate on f by L^
th ^ treating all except the m coordinate ^ as constants, Thus we get a function L (f;x ), Now operate on it by
^^ " th L . treating all but the m-1 coordinate t ..as n .' m-1 m-1
constants. Carry on this process upto an operation by
L . We define L„ to be the resultant operator. Thus n^ N
(1) L^(f;X) = L (L (...L (L^ (f ;xj ix^_^),., ;x2;x ) 1 2 m—1 ra
Now it follows from theorem 1,1,5.1 that if L i
separately have convergence property (L (f;x. ) ->• f(x. ) i
as n. — oo) for the classes
H . (x. ) or uniformly for the classes H „ (S.)), Pi,X(.) 1 H'\±)' ^"'
i=1,2,..., m, then the operators L„ have the convergence
property for the classes H (x) (uniformly for the P 1 A
' m classes H ^ (S ), in the second case), where X and S
P,X^ m ' m m
are the Cartesian products of X/.x, 1=1,2,..., m and
S/.\, 1=1,2,..., m, respectively.
Similarly, it follows from theorem 1.1.5,4 that
if L , 1=1,2,..., m, possess the asymptotic formulae i
of the type (1.1,3.3) given by
62
(2) L^_(r;x.)-f(x.)=-^[f(x.)f^(x.)+f'(x.)i^.(x,) 9i.-i
f"(x.) -x.f^.(x.)j + - ^ — i T2i(^i)-2^i^u(^i)
2 1 + X. ? . (x- ) 1] + o(—7 r) , n.- oo ,
1 01 i''-' ^9.(n.j' ' 1 '
for f £ H^^^ (x.) (or uniformly for f £ H ^ . <a. ,b.> p . , X / . N ^ i' •' p.,X/.s x'1 1 (i) i' (i)
(p. > 2), i=1,2,..., m, then for f £ H^^^ (X) (or •"- P' m
(2) " uniformly for f £ H 4- <S >, in the second case, where '' p,X m ' '
•^' m S is the Cartesian product of the intervals [a.,b.l, m 1- 3_» x J '
i=1,2,,.., m) we have the following asymptotic formula
for the operators L^
m . (3) L^(f;x) -f(x) = 2 ;-f^[foi(-i) f(^)
1=1 ^1^ i'
+ JT, .(x.) - x.'P .(x. )if' (X) + U„.(x.) ' 1x^ i' 1 01^ i') X. 2 ' 2i^ i'
1
P m - 2x:>F, .(x.) + xff .(x.)l f" ] + s o( / \)lI-oo.
1 1l^ l' 1 01^ l' ' X.X.-i . . m.(n.)^ I X 1 = 1 ^1^ 1
V/e notice that terms in the mixed derivatives
f" (kj j; kj j = 1,2,..., ra) are absent from (3). k J
In fact in this case it is possible to show that (3)
holds for a more general class of functions.
1.2 The trigonometric case.
In this section we shall be concerned mainly with
the asymptotic approximation of real or complex valued
2K-peri()dic functions by means of sequences of linear
positive operators. The test functions 1, sin t, cos t,...
63
being bounded on R, in the general setting the results
are applicable only to bounded functions. However, for
many particular operator sequences these results are
helpful in determining the asymptotic approximation of
unbounded functions as well. In the sequel we assume
that if a function f belongs to the domain of definiticn
of an operator L then so does J, the complex conjugate
of f.
1.2.1 Asymptotic formulae for twice differentiable
functions
Regarding an asymptotic formula for tv/ice
differentiable 27t-periodic functions Schurer ([62], p.
22) gave the follov/ing theorem.
THEOREI'I I Let f(t) be an arbitrary hounded 2Ti-periodic
function v/hich is tv/ice differentiable at a point
t=x £ [-''i,'']. Suppose that on the lata i"'/'/-l [-7t,7i] a
sequence (L j (n=1,2,...) of linear positive operators
L applies to all such functions f and possesses the
properties
L (l;x) = 1 + o(-jrUr) n^ ' ^9(n)'
^1 k "" 1 (1) L (sin kt;x) = sin kx + — S — r - + o(—7—r) ^ ' n^ ' ' 9(n) 9(n)^
'^2 k "" 1 L (cos kt;x) = cos kx + —*-?—r- + o(—7—^ ), n ' 9 (n) 9 (n) ''
k=1,2, where 9(n)j^ and 9(n)-* 00 when n-*co.
If there exists a positive integer m such that for the
point X, L applies to (t-x) and there Iiolds
(2) L^((t-x)2^-+2^x) = o ( - ^ ) , (n - ex.),
then we have
64
(3) L^(f;x)-f(x) = [4f(x) {cos x f-| -,(x)
-sin x'P2 ^(x)}-f"(x){cos 2x Yp 2^^'^'^^^^ ^x f a^^^H
: 4?^n) + o ( ^ ) .
(The formula (3) as it is given in [62]contains 2
instead of 4 as the coefficient of f'(n) on the right
hand side. However, it can be verified that the
correct value is 4 as is put in above),
Another theorem in the same direction, but for
a more special class of operators, was given by
Korovkin ([29], p. 99).
THE0REI4 II On the space T of bounded 2K-periodic
functions f integrable on [-7t,7i] let (L j (n=1,2,...)
be a sequence of trigonometric polynomial operators
defined by
n n (4) L^(f;x) = j f(x+t){| + 2 p "" cos kt jdt ,
-71
where (p^ ') (k,n=1,2,...) is a matrix of constants such
that
^ + 2 p^^^ cos kt g 0 ^ k=1 ^
for all t £ [-7i,n] and n=1,2,... and x £ R.
Then in order that for all functions f of T which
are twice differentiable at a point x, the relation
(5) L^(f;x)-f(x) = (l^5^b f"(x) + o(l-p^(")),n ^ o. ,
be valid, a necessary and sufficient condition is that
65
1_p(n)
(6) lira — ^ = 4. n - oo i_p^^ ^
Remark. Let us note that in theorem II the condition
(6) together v/ith the non-negativity of the kernels
1 ^ ( )
•p + 2 P, cos kt, (n=1,2,...), implies that for each k=1
f i x e d k = 1 , 2 , . . . , v/e h a v e
n • ( n ) l i m P^ ^ = 1 ,
n - CO k
and in particular for k=1 this limit relation asserts
that (5) is a meaningful p^symptotic approximation
formula and that the se' uence j L j of operators defines n'
an approximation process for a function f £ T at its
points of continuity.
For a proof 01 this assertion v/e notice that each
of the functions
J (1 + cos kt)!^ + 2 p^"'' cos jt !, (n,k=1,2,...)
j = 1 -^
is non-negative on L-' ,' ] and assumes 8, positive lower
bound on a subset of positive measure. Integrating
these functions beti een the limits -ft and n v/e therefore
have
-1 < P "" < 1 , (n,k=1,2,...).
Thus, in particular the boundedness of the sequence
{p\^^] and (6^ imply that
T • ( - (n 1 , (n), lira i^ + p ^ - - 4 plj 'j = .
n -* 00
Since this is tantamount to
-, . - f • A t-x N
lim 1. "in -2-;x, n -» 00
66
^ ^hich in turn by Gauchy's inequa l i ty im.plies tha t
lim L ( s in —r—> x) = 0 n ^ 2 ' ' n -, oo
and as L (l;x) = 1, (n=1,2,...), for all x, if follov/s
by Korovkin's theorem [30] tnat if f £ T and is
continuous at a point x, then
lim L (f;x) = f(x). n^ ' \ y
n — oo
In particular chuosing f = cos kt, (k=1,2,...) and x-0
the above assertion follows.
As regards theorem I of Schurer, condition (2)
fits very -./ell as long as the operators L explicitly
depend on the functi n values in the interval [-•rt,n]
(in fact it can be shown that in this case (2) is
necessary as v/ell). Iio\."ever, in other cases (2) may be
disadvantageous. To take an . .rtificial example, define
an operator L by L(f;x) = f(x), x ^ 0, x £ [-•n;,!:] and
L(I;O) = f(27i:). Clearly (2) is not satisfied even though
L(f;x) = f(x) for each 2Tt-periodic f. Also the function
(t-x)'"' being an algebraic one, the determination of
L ((t-x)''" ;x) itself may present difficulties, . s v/e
are interested in the apci-oximatio.x --f 27t-pGriodic
functi -ns, it seems reo.sono,ble to replace (2) by another
oultfiblfc condition involving 27:-periodic functions.
w'e aim at establishing the following results:
(l) In theoi-em I relation (l) corresponding to k=2 are
entirely superfluous, that is to say, the asymptotic
formula can be obtained in terms of "f, and ^, . only. 1,1 ^,1
Very recently T-Liamernans [ 75J already proved this fs,ct,
ti.oujjh along different lines, (ii) If, however, (2) is
satisfied and the opex-ators have a meaning for functions
67
(2) of the class Hp ' y(x), m a positive integer, then the
asymptotic formula is valid for functions of this class,
(iii) Condition (2) can be replaced by a necessary and
suffi^KN; condition which can be tested v-/ith the help
of the functions L (sin 2t; n) and L (cos 2t:x) n^ ' n^ '
(n=1,2,..,). Thus the relations (l), if known are
suffient to determine the existence and the form of
the asymptotic formula, (iv) The results that we obtain
generalize theorem II of Koiovkin, which can be obtained
as an easy corollary, \.e also consider the uniform
convergence of the asymptotic formulae in a closed
interval and obtain necessary and sufficient conditions
for this for certain classes of functions. Lastly we
give applications of our results and their genralizations
suitable for applications to the operators v/hich are
defined for functions of several variables.
(') By KX'~J^{x), (x £ R, X ^ R ) , we denote the class c. Tl, A
of a l l bounded functions f defined on X v/hich possess a. 27t-periodic extension f on R \\/hich i s tv/ice
(2) d i f f e r e n t i a b l e at the point x, Q \ <a,b> , 2Tt , A
(-00 < a < b < oo), denotes the class of all bounded
functions f defined on X which possess a 2Tt-periodic
extension f on R which ist-Afice differentiable in an
open interval containing the interval [a,b] such that
the second derivative is continuous at each x £ [a,b] .
In the sequel both f and f are denoted by the common
symbol f,
THEOREI i 1 Let j L , n £ UJ be a class of linear
operators defined on a common domain D(X) ( X ^ R) of
functions into a domain D(X) ( X ^ R) cf functions
and ultimately positive on a set X ^ X. Let 1,
68
sin kt, cos kt, (k=1,2), £ D(X) and let x £ 5', Then in (2)
order that for each f £ D(X) r Q.X ^(x) there holds ^71, A
t h e a s y m p t o t i c r e l a t i o n
( 7 ) L ^ ( f ; x ) - f ( x ) = - ^ [ f ( x ) v ^ ( x ) + f ' ( x ) ( c o s x^^ (x )
- s in X ' i '2(^)l ~ ^ " ( ^ ) {'^°^ ^ "^2^^^ " ^^"- ^ ^ i ( ^ )
-^^-)!^ - ° ^ ^ ) ' where 9(n) ^ 0, 9 ( n ) -' oo as n -* oo , i t i s n e c e s s a r y
and s u f f i c i e n t t h a t t h e r e ho ld t h e c o n d i t i o n s
^ ^ ( x ) ., L ( l ; x ) = 1 + 7—r + o(—7—v)
^^ ' ^ (p (n) ^9 ( n ) ' n ^ ' - ( • ( x )
1 ( 8 ) L ( s i n t ; x ) = s i n x + —T—r + o(—T—v) ^ •' n^ 9 ( n ) 9 ( n } ' ^ ^ ( x ) .,
L ( c o s t ; x ) = cos x + —T—r + o(-7—v—) n^ 9 ( n ) 9 ( n ) '
and
(9) L ^ ( s i n 4 i ^ ; x ) = o ( ^ - ^ ) ,
Further, if [a,b]^ X and if the function
cos xy^/(x) + ;ln x; f. (x) - T, (x) io bounded on [i,b],
then a necei^oary :.nd sr.ffioiant c:adltIo.i that for (2)
each f £ D(X) r^ H\ V < a,b > the relation (7) holds <i , A
uniformly in x £ [a,b] is that (8) and (9) hold
uniformly in x £ [a,b].
Before giving a proof of this theorem, we remark
that Schurer [62] only indicated a proof of theorem I
and that in proving asymptotic approximation formula
for twice differentiable functions for the De La Vallee-
Poussin's integrals, Ilatanson ([51], PP. 212-214)
utilizes the 'relation
69
f(x+2t) = f(x) + f'(x)sin 2t + ^f"(x)sin^ 2t
+ a(x+2t)sin^ 2t
which is inconsistent with the strict 2Ti-periodicity of
f, since putting t = •n:/2 and t = -TC/2 we get
f(x+'n:) = f(x) = f(x-n) which implies that f is TI-
periodic. Thus in the neighbourhood of points t= + TI/2,
in tnis relation, the function a(x+2t) may not be
bounded and therefore the proof requires a modification,
Indeed, if we assume f to be 7i-periodic, Ilatanson's
proof, as it is, is correct.
Proof of theorem 1 Let f £ D(x) r '^[^^{x) , V /riting
(10) f(t) - f(x) - f'(x) sin(t-x) - 2f"(x) sin^ ^
, ,,x . 2 t-x = h^(t) s m - 2 - ,
it is easily verified that v/ith h (x+2mTi)=0, (ra=0,+1,.,),
h (t) is continu us at t=2m7i+x, (m=0,+1,... ), and is
bounded on X. Further h (t) is 2:1-periodic.
Let 6 be a number satisfying 0 < 6 < 2Tt, Then with
\/^ \{t) defined by
0 , if |t-x+2m7t I < 6 for
(11) ^(^ \(t) = I some m=0,+1,...,
1 , othervi/ise,
for an arbitrary e > 0 we have a 6 ( 0 < 6 < 7 ^ such that
(12) |h^(t)| g = + ^'\6,x)^^^' for all ^ £ X,
for some constant M > 0, Clearly
, .. < . 2 t-X/ . 2 5
^(6,x)(^) - ^^" -r^^^^ - . so tha t for a l l n su f f i c i en t l y l a rge
70
(13) L^(h^( t )s in2 ^ ;x) s eL^(sin2 ^ ; x )
+ M L^(sin'^ • ^ ; x ) / s i n ^ 6 / 2 .
I t follows from ( 8 ) - ( l 0 ) and (13) tha t
| L ^ ( f ; x ) - f ( x ) - ^ ^ [ f (x)Y^(x)+f ' (x)Scos x f^(x)
- s i n X ' t '2 (^) ) - f" (x) {cos x1'2(x)+sinx¥ (x ) - ? (x) j ] |
V (x) cosxTp(x)+sinx?'-(x)-Ti (x)
9 X^ - 2 9 ( n )
where lim v (x) = 0. Hence there exxsts a positive n^ / ^ n -» 00
integer II such that for all n g II the left hand side
of this inequality is smaller thane A/9(n) where
A = max (1, I cos x 'H Ax) + sin x Y.(x)-^ (x)| ).
Since £ is arbitrary positive (7) follows. The
necessity of (8)-(9) being a direct verification, this
completes the proof of the first assertion of the
theorem.
To prove the uniformity of (7) in the second
assertion of the theorem, as in the case of similar
results of earlier sections, it is sufficient to show
that M and 6, occuring in (12), can be chosen
independently of x £ [ a,b].
Applying Rolle's theorem to the function
F(0 = f(t) - f(.0 - sin (t-i) f'(0
_ f(t)-f(x)-sin(t-x)f'(x) 2 t:^ . 2 t-x ^^^ 2
sin -g—
where x £ [ a,b] , t £ < a,b> , t / x, it follows that
there exists a lying between t and x such that
71
(14) f(t)-f(x)-sin(t-x)f'(x) ^ j.,(^)^^,(^)^^^ t ^ ^ r-, . c- t-x c,
2 s m p
Applying lemma 1, p. 12, [30], to f", for an arbitrary
E > 0 we can choose a 6 > 0 independent of x such that
(15)
|f"(t) - f"(x)| < J
M. Itan - ^ 1 < f , M, = max |f'(x)| , 1 2 ' 2 ' 1 I - ^ ^ , 1
X £ La,b J whenever | t - x | < 6 f o r a l l x £ [ a , b ] . By (14) and (15)
we have
f ( t ) - f ( x ) - s i n ( t - x ) f ' ( x ) _ ^,,^^-j o • 2 t - x 2 s m —p—
£ I f " ( ^ ) - f " ( x ) | + | f ' ( O l Itan ^ I < e ,
for all t with | t-x | < 6 and x £ [a,b] , where 6 is
independent of x. Having shov/n this, it is clear
that M can also be chosen independently of x. The
second assertion of the theorem then follows.
COROLLARY 1 Let |L , n £ U j be as in theorem 1 and
let (8) be satisfied. Then a necessary and sufficient
condition for (9) to hold is that (7) holds for the
functions f = sin 2t and cos 2t. Also when (s) hold
uniformly in x £ [a,b]^X v/ith the functiwn
cos X ? (x) + sir. -. 'P.(x) - I* (x) being bounded on [a,b]
then a necessary a,nd sufficient condition for (9) to
hold uniformly in x £ [a,b] is that (7) holds uniformly
in x £ [ a,b] for the functions f = sin 2t and cos 2t.
Proof Since the functions sin 2t,
cos 2t £ D(X) r. ^r^\[,x) and L(X) r (i '*„<a,b> both, by ^71,A ^71,A
theorem 1, (9) implies the required satisfaction of (7)
for these functions. The converse proposition follows
72
from the linearity of the situation, t=x being a 4 t-x
fourth order zero of the function sin p •- , This
completes the proof of the corollary.
We omit an analogue of theorem 1.1.3.2 in the
trigonometric case.
Utilising (2) in full and combining the proofs of
theorem 1 and theorem 1.1.3.1, starting from the
relation
(16) f(t)-f(x)-f'(x) sin(t-x)-2f"(x)sin^ ^
, fj.\< - 2 t-x , /, \2m+2 , = h^(t) ! sm -2- + (t-x) } ,
we arrive at the following result.
THEOREM 2 Let { L , n £ Uj be a class of linear
operators defined on a common domain D(X) (X £^R) of
functions into a domain D(X) ( X ^ R ) of functions and
ultimately positive on a set 5?'^X. Let 1,sin t, cos t,
t,t , ..., t £ D ( X ) , where m is a positive integer,
and let x £ "X. Then in order that for each
f £ D(X) r ^2^1.2 X^^^ ^^^ relation (7) holds, it is
necessary and sufficient that (2) and (8) hold,
Further, assuming that the function
cos X p(x) + sin X ¥ (x) - T (x) is bounded on
[a,b]^X, a necessary and sufficient condition for (7)
to hold uniformly in x £ [a,b] for each
f £ D(X) r, H^^lg X <^»^> ^^ ^^^^^ ^2) and (8) hold
uniformly in x £ [a,b].
It is clear that the result of theorem 2 can be (2) (2)
extended to classes H^ -;,(x) and H^ i <a,b>(m > 2 and a m,X m,X
positive number) if we reformulate the theorem in terras
of the expression L (|t-x| ;x) in the manner of theorem
1.1.3.1.
73
Next v/e consider a class of sequences of linear
positive operators generated in the following v/ay,
Let (p, ) and (6^ '), n,k=1,2,..., be two oo x oo
raatrices of real numbers satisfying the follov/ing
properties
(i) ^ + 2 (pf' -'cos kt + bl^^ sin kt) is ^ k=1 ' ^
uniformily convergent on the interval [-TI, TC] for each
n=1,2,... .
(ii) 4- + 2 {ry-' cos kt + b}^^ sin kt) S 0,t £ [-7i,Tt], ^ k=1 ^ ^
n=1,2,... ,
Along the lines of the remark to theorem II, let
us note that (i) and (ii) imply that -1 < p^. ,
6 " ^ < 1 for n,k=1,2
Putting
U^(t) = - + 2 (p^^^ cos kt + b^' sin kt), k=1
n=1,2,..., on the space T (as defined in theorem II) of
functions define a sequence JL j(n=1,2,...) of
operators oj
(7) L (f;x) = 7 f f(x+t) u^(t)dt ,
-oo < X < oo ; n = 1 , 2 , . . . .
Applying Korovkin's remark to Theorem 4, p. 18,
[ 30] , v/e have
THEOREM 3 Let {L^j(n=1,2,...) be the sequence of
operators defined in (17) such that (i) and (ii) hold.
Then in order that for each f £ T which is continuous
74
a t '- p o i n t X £ R v/e h a v e
( 1 8 ) l i m L ^ ( i ; x ) = f ( x ) , n -* CO
it is necessary and sufficient that
(19) lira p ^ ^ = 1. n — "
Further, (19) is also a necessary and sufficient
Condition th-.,t for each function f £ T v/hich is
continuous at each point of a compact set S R, (l8)
holds uniformly in x £ S.
COROLLARY 1 If (l9) holds, then for each fixed k
v/e iiave
(20) lim '.<i^^ = 0 , (k=1,2,...), n -> CD
and
(21) lira i^l""^ = 1 , (k=2,3,...). n - c
Proof Let (l9) hold. By theorem 3 for each fixed
k(k=1,2,...) we have
lim L (sin kt;0) = 0 n^ / n -' vo
and
lim L (cos kt;O) = 1, n^ y
n -- c-
As L (sin kt;0) " -)f ^ and L (cos kt;0) = f^\ (20)
and (21) follov/, Tnis completes the proof of the
corollary,
In the next theorem v/e determine a necessary and
sufficient condition for the existence of an asymptotic
formula for twice differentiable functions,
TIir-ORFlI 4 Let L (n=1,2,...) be the sequence of
operators defined in (l7) such that (i) and (ii) hold.
Then for each f ' T v/hich is tv'/ice differentiable at a
75
point X G R, there holds
(22) L^( f ;x ) - f (x ) = •.5'^)f'(x) + ( l - p 5 " ) ) f " ( x ) + o ( l - p 5 ' ' ) ) ,
as n -* oo if and only if
. (n) 1-Pp '
n -> oo 1 _ p; '
Further, (23) is also a necessary and sufficient
condition that for each f £ T, which is twice
differentiable in an open interval containing the (2)
closed interval [a,b] (-or, < a < b < 00) with f (x)
continuous at each x £ [a,b], (22) holds uniformly in
X £ [a,b] ,
Note As in the remark to theorem II, here also we find
that (23) implies (19). Thus by the corollary to
theorem 3 it follows that (22) is a meaningful
asymptotic approximation formula.
Proof of theorem 4» We have for all x
L^(i;x) = 1 T / • _L \ (n) . , (n) L (sin t;x) = p^ 'sxn x + 6 'cos x L (cos t;x) = p!; -'cos x - b\ ' sin x
l-o '" , / . 2 t-x N 1 n^^^^ ~T"' ^' " — 2
and L^(sin^ ^f^jx) = J (| + - 2p ^ ) ,
Moreover,
L^(sin(t-x);x) = b^f^ .
Let us take 9(n) = (l-pij ) . Then condition (9) of
theorem 1 becomes
| - ^ ^ - 2 p S - ) = o(l-pW) .
76
Since -1 < plj < 1, dividing by l-plj^'^ t h i s i s seen
to be equivalent to (23) . As the terms
{cos X .' ( x ) - s i n J (x)j/c[)(n) and
-{cos X ;'2(x) + s in x J^ix) - ^(x) / (n) in (7) are
j u s t asymptotic evaluat ions of L ( s i n ( t - x ) ; x ) and
L (2 sin ~^i x) r e spec t i ve ly , the formula (7) i s
rc.'-dily seen to be equivalent to (22) in t h i s case .
Having observed t h i s theorem 4 follows from theorem 1,
Theorem I I of Korovkin i s a p a r t i c u l a r case of
tneorem 4 v/hen ' ^^ = 0 (n ,k=1 ,2 , . . . ) and -J"^ = 0
( k = n + 1 , n + 2 , . . . ; n = 1 , 2 , . . . ) ,
Applications of above theorems eas i ly give
asymptotic formulae for the operators A ( f ; x ) of
Korovkin ( 30 , p . 74)» the De La Val lee-Pouss in ' s
operators V ( f ; x ) ( 51 »[62 ) , the Jackson 's operators
Lo o (e .g . r62l) and t h e i r genera l i za t ions L 2n-2 ^ " - J' ° np-p s tudied by Schurer ( [ 6 2 ] , chapter 3) and o the r s .
Jn a l l these cases the asymptotic formulae hold (2)
unif irmly in x £ [a,b] for f £ tA ^ <a,b> and integrable
on [-71, Ti], where [a,b] is any bounded :ind closed
interval on the real line. The Pejer operators L _.,([62])
and the ? dsson operators L (f;x) ([30],p, 192) do not
satisfy the conditi m (9) 01 theorom 1 (or the
condition (6) or (23) of theorems II :nd 5, respectivelj^
and therefore do not possess such asymptotic formulae.
A convenient reference in v/hich evaluations (1) for
most of these operators can be found is [62].
Before proceeding to the next section we give one
more result for the determination of an asymptotic
formula. This result depends on the fact that for 2 2
k=^,3,..> the function cos kt - k cos t + k - 1 has a fourth order zero at t=0 and is positive at all
77
other points of the interval [-71,7:],
THBOREI-I 5 Let { L , n £ Uj be a class of linear operators
defined on a common domain D(-X) (X S- R ) of functionc
i-ito a domain D(X)(X ^ R ) of functions and ultimately
positive on a set X 3=X, Let D(X; contain all trigono
metric polynomials. Let x £ X, Then in order that for (2)
each f £ D(X) ^ Q^ v(^) there holds the asymptotic £- 71, A
relation
(24) L^(f;x) - f(x) = f(x) { L^(l;x)-1 j
+ f(x) L^(sin(t-x);x)+2f"(x)L^(sin^ ^ ; x )
+ o(L^ (sin - ^ ;x)), n - c. ,
it is necessary and sufficient that for some k
(k=2,3,...) we have
. / . 2 k(t-x) ^ L (sxn - ^ <- ; x) p
(25) l i - ; . , 2 t - x , — - ^ ' n - 03 L (sxn —^T" 5 x) n^ 2 ' '
Further, a necessary and sufficient condition (2) 2 " 2
the relation (24) holds uniformly in x £ a,b is that
(25), for some k(k=2,3,...), holds uniformly in
X - _ a,b ,
Proof From (25) it follows that
, / . 2 k(t-a) N , 2 ., , . 2 t-x ^ L^(sxn -^2 ^?^) = k L * ^ ^ ~ 2 ~ ''^'
+ o(L^(sin -~; x)), n - co
which is equivalent to
(26) L (cos k ( t - x ) - k ' ^ c o s ( t - x ) + k 2 - 1 ; x )
= o(L„ ( s i n - ^ ; x ) ) , n - oo.
t h a t f o r each f £ D(x) r^QlX ' < a , b > ( ' a , b ' ^ X )
78
If f £ D(X) r Q^2) (x) , for all t £ X 271,X^ '
we have
(27) f(t)-f(x)-f'(x)sin (t-x)-2f"(x)sin2 ^
, / N . 2 t-x = h^(t) sxn -g- ,
where h (t) is such that given an arbitrary e > 0
there exists a 6 > 0 and a positive number M. , say,
such that for all t £ X there holds
/'nn^ ll /j_\l • 2 t—X ^ . 2 t—X (28) h (t) sxn -T;-^ e sxn —r-\ / I ^\ /1 2 2
2 2 + M jcos k(t-x)-k cos(t-x)+k -1j.
A proof of (28) utilizes L'Hospital's rule and the 2 2
property of the function cos kt-k cos t+k -1 stated
at the beginning of theorem 5. Using (26)-(28) the proof
of (24) follows, (2)
In case f £ D(X) n. KX „ <a,b>, 6 and M in above ^ 7t, A 0
can be chosen to be independent of x £ [a,b]. Then
uniformity .of (25) for x £ [ a,b] implies the uniformity
of (26) for x£ [a,b] v/hich in conjunction with (27) and
(28) shov/s that (24) holds uniform-ly in x £ [ a,b] ,
Taking f(t) = sin ^p~—^ the necessity parts of
the theorem are evident. This completes the proof of
the theorem.
1,2.2 Generalizations for functions of several
variables.
V/e state without proofs some generalizations of
results of section 1,2,1 to the case of operators
defined for functions of finitely many variables. The
proofs may be given along the lines parallel to those
in [59], [62] and in the previous results of this
thesis. We follow the conventions and notations of
79
sections 1.1,5 and 1,2.1,
By Q i X ( )' ^^ ^ \ » \ - \ ) ' '' ' ''° ^ * ^ ' m
class of all bounded functions f(?,. ,..., ^ ), defined
on X , which with an extension on R are 27i -periodic m m -
in each of their arg-uments and are tv/ice differentiable
(in the sense of section 1.1.5; at the point X.
Qoiv < S > , (S ,X R ), denotes the class of all 27f X ra ' ra' m — m^'
' m bounded functions f(E^,...,r ), defined on X , v/hich
1 ' m'' ra' with an extension on R a-re 2it-periodic in each of
m ^ their arguments and are uniforraly tv/ice differentiable on S (in the sense of section 1.1.5).
m THEOREM 1 Let { L , N £ U jbe a class of linear
operators defined on a common domain D(X )(X C: R ) of ^ m' m — m'
functions into a domain D(X ) (X "S:R ) of functions ^ ra^ m — m'
and ultimately positive on a set X '—.X , Let 1,sin t., "" ^ m m ' J' cos t., sin 2t., cos 2t., sin t. sin t, , sin t. cos t, ,
O' j' j' 0 k' 0 k' cos t. sin t, , cos t. cos t, (j / k; j,k=1,..., ra)£D(X ).
J k' J ^ .^ , , ... , - , , - . ^
id only if the op
that at a point X £ X
If and only if the operators L.. possess the property
m „, m ^-- "^
(1)
L„(1;X) = 1 + 2 - ^ r + 2 o(— ] r) , N • ., 9 • (n. ) . . ^9. (n. )^ '
1 = 1 ^x^ x^ x = 1 ^x^ J.'
L,.(sin t.;X) = sin x. + 2 — ' ^ ^ \ + 2 o(—7 r),
m ?2ii '^ 1 L„(cos t.;X) = cos x.+ 2 — j r + 2 o( 7 r) , N y J ^^^ 9i(n^) ^^^ >^(n_^)^'
ra Y, .,. m L„(sin (t.-x.)sin(t, -X, );X)= 2 ''y"\+ 2 o ( — r ^ ) ,
(j / k; j,k=1,,.., m),
and
80
t.-x. m . (2) L (sin^ ly-^ ; X) = 2 o(—-J—r), (j = 1,,,,,m),
as N-* oo , where 9 . (n. ) j 0, 9. (n. ) — 00 , n. — 00 , and
?'s are functions of X, then for each
f £ D(X ) r QA^iv (X) there holds ' m
m ( 3 ) L.,(f;X) - f ( x ) = 2 —7 r [ T . f + ^ ^ N ' ^ ^ ^ ^^^ (f^{n^) '- ox
m 2 (^-, . . cos X. - Yo • • s i n x . ) f '
j ^ 1 101 J 2jx 0^ X.
-, m ra + r- 2 ? , . , . f" + 2 ( f . - 'P^ . . cos X.
2 k , j = 1 5jkx Xj^x. .^^^ ox 2jx 0 k ^ j
m - " i j i ^ '^^a^^x .x . l - . ^^^ ° ( ^ T ^ ) ' ( ^ — ) .
where va lue s of t h e f u n c t i o n s o c c u r i n g i n (3 ) a r e t a k e n
a t t he p o i n t X,
F u r t h e r , l e t S be a bounded s u b s e t of 5 and ' m ra
assume that the functions f . - f„.. cos x. - v-.. sin x. ox 2ox J Mjx J
(i,j=1,...,m) are bounded on S , Then for each (2) "
f £ D(X ) r Q, i V (S ), (3) holds uniformly in x £ S m 2n*,X m m ' m if and only if (1) and (2) hold uniformly in X £ S .
Also, the above statements remain true when (2) (2 )
i s r e p l a c e d by ( l . 1 , 5 . 1 l ) a l o n g w i t h Ol „L y (X) and c- 71 .A
(2) (2) ™ Q^^; X ' m' being replaced by H '' (x) and (2)' ^
H ; .;: < S >(p > 2, j = 1,...,m) respectively. p,A^ m J
COROLLARY 1 In the statement of theorem 1, if (1) hold,
then (2) are equivalent to the condition that (3) holds
for the functions sin 2t. and cos 2t., (j=1,...,m). In
81
the uniformity part of theorem 1, if (l) hold
uniformly in X £ S , then the unifornity of (2) is
equivalent to the uniformity of (3) for the functions
sin 2 t. and cos 2t., (j=1,...,m), for X £ S .
An analogue and generalization of theorem 1.2.1.4
is the following (n.) (n )
THEOREM 2 Let (p , " ), (&/ ), (n . ,k=1, 2,... ;
j = 1,..., m), be 2m, 00 x 00 matrices of real numbers
satisfying the properties
-I °° (n.) (n ) (i) ^ + 2 (pi ^ ) cos kt + 6, ^ sin kt) is uniformly
^ k=1 ^ '
convergent on the interval [-71,71:] for n. = 1,2,...; J
j = 1 , 2 , . . . , m,
1 °° "^^i) "^" i^ ( i i ) T5- + 2 (pi cos k t + 6, 'J s i n k t ) = 0, t £ [-7i ,7.],
2 k=1 ^ ^
:iie
n , = 1 , 2 , . . . ; j = 1 , 2 , . . . , m. On t h e space T of a l l
f u n c t i o n s f ( t - , . . . , t ) , 27 i -por iod ic i n each of 1
argument t . , j = 1 , . . . , m) , bounded i n R and i n t e g r a b l e
o n [ - 7 i , 7 i ] v/ithi r e s p e c t t c t . , ( j = 1 , . . . , m ) , v/hile o the r
arQ-'am.ents assume a r b i t r a r y but f i x e d v a l u e s , d e f i n e
t he o p e r a t o r s L,. by
7: Ti;
(4) Ly(f ;X) - -^ r . . . [ f(X+T) .
- 7 1 - 7 1
ra 00 ( n . ) ^ V
n i j + 2 ( p ^ cos k t + 6|.^^Gin k t . ) j j = 1 "^ k=1 ^ J - J
d t . . . . d t , 1 m'
X £ R ; n , n , . . . , n =^^1,2, . . . , '..iiere X+T =
( x . + t . , . . . , A +t ) . Ihen i n orJ.. • t i n t x. .:• each ^ 1 I ' ' n v/
82
i £ T ^ 1; i. -, (x), (X £ R ), there holds the relation m 71*,iv ^ " ^ n "
m m (n ) (n )
(;) L,(f;x)-i(x) = 2 &-, 5^ f" (X) j,k=1 j k
' ^ k m (n.) m (n.)
+ 2 6 J f^ (X) + 2 (1-p^ J )f^ (X) + j=1 j j=1 j j
m (n ) + 2 o(l-p J ), j=1
(E -• oo), a necessary and sufficient condition is that
(n,) J
,,, '-P2
\o) li-i — [ v r y = 4 , j=1,2,..., m.
Purthor, (6) is also a necessary and sufficient
condition tnat (5) holds unifoj>mly in X £ S for (2) ^
occn f £ T r\ <X „ n <S > v/here S is an arbitrary m 2 71 *, n m. m
ri
subet wi R . n
Remark 1. In theurera 2 -e note that (i) and (ii) imply (n ) (n.)
thct -1< Pv ^ ,6k ' < 1 i^°^"j'^=1'2^A-)' (n.) j = 1,2,..., m. Tnus the expressions (l-p^ )/'\'i-p^ ),
j = 1,L,...,m, in (6) 8,re meaningful. Moreover, v/e observe
that (6) hold only if ("i) xim pj "J = 1, j = 1,2,..., m, n .-» 00
(nj In v,-hic";i case automatically lim 6., = 0 ,
n . -• 00 J
j-^1,2,... ,m, showing that (5) is o- meaningful asymptotic
relati .m.
Remark 2. Let jpi j,{6, j, (l:=1, 2,...), be tv/o sequences
o'' real numbers such th t tne secies
83
(7) S(t) = • + 2 (pi COS kt + 6, sin kt) ^ k=1 ^ ^
is uniformly convergent on [-TI, 7t]. Further assume that
(8) S(t) go, t £ [-71:,T:].
Then there holds
1-Pp
To prove (9), we observe that for all real t
(10) sin' I ^ sin^ I .
Since there must exist a set A [ -7i;,7i] of positive
measure on which strict inequality holds both in (8) -1
and (10), multiplying (l0) by — S(t) and integrating
from -K to 7t we have
|(|-^T-2Pi) =f (1-Pi) where a is some positive constant.
1-Pp
,4- = 4(1-«)
It follows that
from which (9) is obvious,
Using (9) it follov/s that the conditions (6) are
equivalent to each of the following two conditions
^\) m l-pp
(11 ) lim 2 7-—Y = 4m, II - oo k=1 , ' k
1-p,
^ 1~Pp p„ (12) lira n 7—r = 2 ^ ,
N - oo k=1 , " k^ 1-P1
Remark 3 Inequality (9) can be generalized to the
84
follov/ing inequa l i t y
^ ' '"Pk 2 (13) ^ — ^ < k^ , k = 2 , 3 , . . . ,
Pi
under the same assumptions (i.e. S(t) is uniformly
convergent on [-71,71] and that (S) holds).
A proof of (13) is as follov/s.
From the elementary inequality (k=2,3,...)
(14) cos kt - k^cos t + k -1 a 0, t £ [-TI:,7I],
v/hich v/e have used in the proof of theorem 1.2.1.5,
and the fact that strict inequalities hold in (8) and
(14) n a subset of [-7i,n] of positive measure, as
before, \/e have the inequality
Pk - ^^Pl + ^^-^ > °
which is tantamount to (13). Incidentally, (13)
improves on the following inequality of Stark [ 73],
p. 24,
1-P-l 2
In the light of theorem 1.2.1.4 it follows from
(9) and (13) that the operator sequences (I.2.I.I7) in
theorem 1.2.1.4 possessing asymptotic formulae for
tv/ice differentiable functions are precisely those
which attain the upper bounds in (9) and (13) in a
limiting sense.
V/e omit an obvious generalizatio.i of theorem
1.2,1.5 for the many vari;.',ble case,
As Indicated in section 1.1.7, we can construct
operators for functions of several variables v/ith the
85
help of those for functions of fev/er variables. Results
of this section are readily applicable in studying their
approximation properties. In fact the operators in (4)
are constructed precisely in the same manner.
Also we remark that some of the results of earlier
sections, though intended for algebraic functions, may
in cases furnish us with asymptotic formulae for periodic
functions more conveniently than the results of this or
the preceding section (e.g. for the operators L ,
p > 1, Schurer utilizes Theorem 2, [62] to obtain the
asymptotic formulae).
1.3 Approximation of functions of an exponential
growth.
In this section we require the following extra
notations:
E „ : class of all real (or complex) valued functions cc, A
f(t) defined on X ( S R ) , to each of v/hich there exists
a positive constant A such that |f(t)| < A e ' ' for all
t £ X. Here a is assumed to be a fixed positive real
number. E „(x): class of all f £ E „ which v/ith an a,X^ (X,X
extension on R are continuous at the point x(£ R ) , (2)
E^ -^(X): class of all f £ E which with an extension 0£,A a,A
on R are twice differentiable at the point x.
Let S be a subset of R. By E -^{S) v/e denote the ex, A
class of all f £ S ^ which v/ith an extension on R are a,X
continuous at each point of S. Also let <a,b> denote
some open interval (c,d) containing the closed interval
[a,b]. By E^ I <a,b> we denote the class of all f £ E „ ^ -' a,X a,X
which with an extension on R are twice differentiable
at each point of <a,b> v/ith the second derivative
86
continuous at each point of [a,b],
1,3.1 Single variable case
After having considered the periodic case in
section 1.2.1, proofs of various convergence results
for functions of exponential classes follov/ nearly
the same pattern v/ith the difference that here v/e use
hyperbolic functions instead of circular ones.
Following is the basic convergence theorem for
the classes E . (x) and E ^(S). (X fX 0£ , A
THEOREI'I 1, Let j L , n £ U j be a class of linear
operators defined on a common domain D(X) (X cr R) of
functions into a domain D ( X ) ( X S ' R ) of functions and
ultimately positive on a set X .S-X, Let a > 0 and 1,
e"^, e"""* £ D ( X ) . Let x £ X, Then, if and only if the
operators L possess the property that in
L (l;x) = 1 + a (x),
(1) L^(e'^;x) = e"^ +P^(x), and
T / -at >, -ax , / >, L (e ;x) = e + y ix) n^ ' ' 'n^ '
there hold
(2) lim a (x) = lim (3 (x) = lim y (x) = 0, n — oo n-'oo n-*oo
then for all f £ D(x) r E „(x)
(3) lira L^(f;x) = f(x), n — oo
Furtiier, let S S. X be a compact set. Then for each
f £ L(X) E „(S), (3) holds uniformly in x £ S if and
only if (2) hold uniformly in x £ S,
A proof of theorem 1 is obtained by using the
following: v/hen f(t) £ E y{x), there exists to each a, A
87
e > 0 a positive number A, depending also on x, but not
on t, such that for all t £ X
(4) |f(t)-f(x) I < e + A sinh^ ^ ^ % ^ ,
In case f £ E ^(S) (S compact), A can be chosen a, A
independently of x £ S.
A criteria, for asymptotic formula for the classes (2) (2)
E^ ^(x) and E ) <a,b> (p > O) is given in the following
theorem, V/e take (3 to be of the form ma v/here m is
a positive integer (m = 2),
THEOREI'I 2 Let {L , n £ U j be a class of linear operators
defined on a common domain L(X)(X R) of functions into
a domain D ( X ) ( X S R ) of functions and ultimately
positive on a set X C x. Let a > 0 and assume that the
functions 1, c " (k=+1, +2,..., +m) £ D(X) where "n g 2
is a positive integer. Let x £ }<f. If and only if the operators L satisfy the conditions
n 'i'o(x) .,
? (x) (5) L^(e ;x) = e + - ^ ^ + o ( ^ ,
W (x) T / -gt \ -ax , -g^ /'_1_\ L (e ;x) = e + —TTTT + ^(TT;^) » n'
and •*" 9(n) " °>(n)'
(6) L^(sinh ^ a(-=^);x) = o ( - ^ ) ,
as n -• 00 , where 9(n) / 0, 9 (n) -• 00 as n — 00, then
for each f £ D(X) ,- E^ ^(x) there holds the asymptotic ma,X^ ' •' ^
formula
(7) L^(f;x) - f(x) = —^ [2a2f(x)? (x) 2a 9(n)
+ af'(x)i e-"\(x) - e"^^_^(x)j
88
+ f"(x) {e-«"Y^(x) + e " X j x ) - 2Y^(x) j]
Further, a necessary and sufficient condition that
for each f £ D(X) r^ E^^^ <a,b> ([a,b] ^X), (7) holds
uniformly in x £ [a,b] (we assume that the function
e""^^ (x) + e"' ? (x)-2T (x) remains bounded on [a,b]). a —a o L » J'
A proof of theorem 2 is based on the relation
/Q^ s.f^\ fl \ , f(x) . , /, s , 2f"(x) . ,2 a(t-x) (8) f(t) = f(x) + —^—^ sxnh a(t-x) + ^—'- sxnh ^ p ' a
, 1. /'x ( • 1.2 a (t-x) , . , 2m a (t-x), + h (t)( sxnh —^ '- + sxnh —^—5—'•j
(2) where with f £ E^ Y(^) and h (x) = 0, the function
g'
h (t) has properties similar to the ones encountered in
the algebraic and the periodic case. We omit the details.
For the uniformity part we apply Rolle's theorem
to the function
P(0 = f(t) - fix) - ^ ^ sinh a(t-0
f(t)-f(x)- ^ 1 1 ^ sinh a(t-x) P ." . '- sinh^ ^^j^
with X £ [a,b], t £ <a,b> (t ^ x) and f £ E^^'^^<a,b>, a
Here we are taking f as extended on the whole of R and
so t is not necessarily restricted to X only and
assumes values on R. A restriction of t to X will be
required only in the final steps. It follows that for
some % lying between t and x
f(t)-f(x)- ^'^^^ sinh g(t-x)
2 -,,2 0 —^ sxnh — 2 g
^ ^ ^ = f"(0+af'(£)°°^^°/^"^h^ (t-x) ^ u;+ai U^si^^^,^)
89
Hence we have
h,(t) = ^ [f"(o-f(x)+af.(o :°:iiiii-)']. a
(. . , 2ra-2 a ( t - x ) )-1 , i1 + sxnh ^ 2 ' ]
The function cosh a(t-^)-1 has a second order zero at
t=^. Hence applying Lemma 1, p. 12 [30], to f", it is
clear that given an arbitrary e > 0 v\re can choose
a 6 > 0, independent of x £ [a,b] such that with t
restricted to X
|h^(t)| < e , if |t-x| < 6 , X £ [a,b], t £ <a,b>,
Having chosen 6 in above manner, it is clear from the
expression
f(t)-f(x)- ^ ^ s i n h a(t-x)- ^ ^ ^ sinh^^a^^
h (t) X
a
3inh2 - ^ + sinh2- ^ % ^
that there exists a positive number M, independent of
X £ [a,b], such that |h (t)| < M whenever |t-x| g 6 and
t £ X. Now, b and M having been chosen independently of
X £ [a,b_|, the uniformity part follov/s. The necessity of
various conditions is obvious,
COROLLARY 1 In the statement of theorem 2 the conditions
(5)-(6) are equivalent to the requirement that (7) holds (Ykt
for the functions f = 1, e (k= +1,..., +m). Further,
assuming that the function e""^? (x) + e"^? (x) - 2? (x) a -a o^
is bounded on [a,b], the uniformity of (5)-(6) for
X £ [a,b] is equivalent to the uniformity of (7) for the
above mentioned functi. ns. Assuming that for certain values of the parameter p
an evaluation of L (e' ;x) can easily be obtained, the
following corollary gives tlie most convenient method of
90
testing and an immediate determination of the asymptotic (2)
formula for functions of the classes E^ \r(x) and (p) ma,X'
E^^\<a,b>, ma,X
COROLLARY 2. Let jL , n £ Uj be a class of linear
operators defined on a common domain D(X) (X R) of
functions into a domain D(X)(X R ) of functions and
ultimately positive on X £ X, Let a > 0 and m g 2 be a ka positive integer and assume that the functions 1, e
(k=+1,...,+m) £ D ( X ) , Let x £ "X, A necessary and
sufficient criteria for the existence of an asymptotic
relation of the type (7) for all f £ D(X) r> E^^\(x) is ma, A
that
(a) L (e^Sx) = eP^ + e^^ ^^^ ''H^'^^ " g^v(x) ^ n^ ' ' 9(n)
for p= ka; k = 0,+1,,.. ,+m, where x(x), |i(x) and v(x) are
functions in x independent of p, The asymptotic formula
is given by
(10) L^(f;x) - f(x) = [f(x)x(x) + f'(x)j,(x)
+ f"(x)v(x)] + o(^l^). n -«, ,
Further, assuming that v(x) is bounded on [a,b] c. X,
a necessary and sufficient condition that (IO) holds uni-(2)
formly in xe[a,b]for each f£D(X)r E < ,ii>is that (9)holds
uniformly in x £ [a,b] for the values of p specified
above.
1.3.2. Generalizations for functions of several variables
In this section v/e give some generalizations of the
results of section 1.3.1. Proofs of the results are
omitted, which, however, can be obtained along the lines
91
indicated in section 1.1.5. In addition to the
conventions and notations of section 1.1.5 we record
the following.
By a we denote the m-tuple (a-,ap,...,a ) and we
take a.(i=1,...,m) to be positive numbers, E : class m
of all real (or complex) valued functions
f(t-,to,..., t ) defined on X («S.R ) to each of which ^ 1 ' 2' ' m^ m^ m" there exists a positive number A such that
m a. 11. I |f(t.,t t )| < A 2 e "- ^
i = 1
for all (t.,t„,...,t ) £ X . ^ 1' 2' ' m' m
E „ (x): class of all f £ E „ which v/ith an extension a,X a,X on R are continuous at the point X (£ R ). /p-vm ^ m'
E i (x): class of all f £ E ^ which with an extension a,X ' a,X ' m m on R are twice d i f f e r e n t i a b l e ( in the sense of sect ion m ^ 1.1.5) a t the point X.
Let S be a subset of R . By E „ (S ) v/e denote the m m " a,X m'
' m class of all f £ E ^ which with an extension on R
a,X ra m .2)
are continuous at each point of S . Also by E^ J. (S ) m " a,X m'
' m we denote the class of all f £ E ^ which with an
a,X ' m
extension on R are uniformly twice differentiable on S m ' m
(in the sense of section 1.1»5).
THE OREM 1 Let iLvrjN £ U j be a class of linear operators
defined on a comnon domain L(x)(X ^ R ) of functions m m'
in to a domain D(X ) of functions and u l t imate ly pos i t ive TTl 4 - 0 .
~ ^ " o^i^'i ~ g i ' ' i on a set X •:= X . Let the functions 1, e " " , e " " m m ' ' ( j = 1 ,2 , . . . ,m) £ D(X) and l e t X £ X . i /r i t ing m
Lj^(l;X) = 1 + a^^iX),
92
m a. t. m a .x. L ( 2 e ^ ;X) = 2 e +p (X), and
j = 1 j = 1 ^ '
m -a . t. m -a . x . L ( e J J;X) = 2 e " + v„ „(X) ,
j=1 j=1 ^'"
if and only if there hold
(2) lim a^ix) = lim p (x) = lim v (X) = 0 JN-»co XJ-.00 ' N-*oo '
t hen f o r a l l f £ D(X ) r^ E „ (x ) v/e have ^ m' a,X ^ ' ' m
(3 ) lira L ( f ; X ) = f ( X ) . N - oo
Further, let S ^ X be a compact set. Then for ' m m ^ each f £ D(X ) / E ^ (S ), (3) holds uniformly in
m X £ S if and only if (2) hold uniformly in X £ S .
THEOREM _2 Let {l TjN £ U j be a class of linear operators
defined on a common domain D(X ) (X "=. R ) of functions m' ^ m m
i n t o a domain L ( X ) ( X " ^ R ) of f u n c t i o n s and m m m u l t i m a t e l y p o s i t i v e on a s e t X ^ X . Let t h e f u n c t i o n s
,1 1 , 4. , X m m
+ k . a . t . ± g - t . + a , t ,
1, e J ^ ^ e J J ^ ^ ( j / k ) , (k = 1 , 2 , . . . , p . ;
j , k = 1 , 2 , . . . , m) £ D(X ) v/here p . g 2 , j = 1 , 2 , . . . , m, a r e
p o s i t i v e i n t e g e r s . Le t X £ X . Then i f and only i f
t h e r e ho ld ra ? . m
1 = 1 ^ 1 ^ l ' 1 = 1 ^ 1 ^ 1 ^
a . t . a . x . m T,, . . m
1 = 1 ^ 1 ^ l ' X = 1 ^ 1 ^ 1 ^
- a . t . - a . x . m f m ^
L^(e ^•^;X)=e ^ ^ + _. 2 ^ . _, o(-irhn^' ^ / \ + 2 O 7 y, . . cp. (n . ) . ^ ^cp. (n . j • 1=1 ^ 1 ^ 1^ 1=1 ^ 1 ^ 1^
(5 ) Lj j (s inh aj^(tj^-xj^)sinh a ^ ( t ^ - x ^ ) ; X )
93
m ? ra
1=1 ^ 1 ^ 1^ 1=1 ^ x ^ x^ and
2 p . a . ( t . - X . ) m
(6) L (sinh J - i ' ' ) = .\ °^v:ijr)^ ' 1 = 1 ^X^ l '
a s H - oo , w h e r e j , k = 1 , 2 m; 9 j_ (n^ ) / 0 , 9 ^ ( n ^ ) - «=
(7)
a s n . -» oo , 1 = 1 , 2 , . . . , m , t h e n f o r e a c h
f £ L(X^) E^2^^ ( ) ^p^ (p^^^^ Ppap,.... P^aJ) • ' m
there holds the relation m -
L„(f;X) - f(x) = 2 / \ [Y^. f
m . -a.x. a.x. + 2 ^ (e ^ - ip, .. - e ^ - f .. )f' j=1 2«j 1J- 2ox> X.
m + 2 -^r-^— T,, .. f" k,j=i 2«j,gj 5kjx xj x.
k / j m ^ -a.x. a.x.
+ 2 — 5 — (e '^ ^ -a... + e -^ • '? ..-4Y • )f" ] . . . 2 1ix 2ii 01' x.x.-' J = 1 4aj ' ^ J J
m
1 = 1 ^ 1 ^ 1 ^
where values of various functions occuring in (5) and (7) are taken at the point X.
Further, assuming that the functions -a.x. a.x. e ^ J V,, . + e ^ ^Ypji - 4^Q. (i,j = 1,..., m) are
bounded on S C_ X , where S is a bounded set, a m •— m' m ' necessary and sufficient condition for (7) to hold
unif.-rmly in X £ S for each f £ D ( X ) r^ 'E^'^K. (S ) that ^ m ^ m^ pa,-v m' ' ra
(4)-(6) hold uniformly in X £ S .
COROLLARY 1 In the statement of theorem 2 assume that
(4)-(5) hold. Then a necessary and sufiieint conxition
94
for (6) to hold is that (7) holds for the functions a .k.t. e " • ', j = 1,..., m; k = +2,..., +p . In case (4)-(5)
J J
hold uniformly in X £ S and the functions -a.x. a -x. e ^ " f, .. + e ' ' ¥„..-4f • (i, j = 1 ,... ,m) are bounded Iji 2ji ox '' ' ' ' on S , a necessary and sufficient condition for (6) to
m' " ^ ' hold uniformly in X £ S is that (7) holds uniformly in
m X £ S for the functions mentioned above. m COROLLARY 2. Let f L.,, II £ U j be a class of linear
operators defined on a common dom.ain D(X )(X S . H ) ^ ^ m'^ m m
of functions into a domain D(X ) of functions and ultimately positive on a set X X . Let the functions
+k .a .t. +a .t .4a, t, 1, e ' ^ \ e ^ ^ ^ (j / k ) , (k.=1,2,...,p.(g 2);
j , k = 1 ,2 , . . . , m) £ L(X ) and l e t X £ "X . Then the
conditions
(8) L„ (sinh a, (t, -x, ) sinh a . ( t . -x . ) ; X) \ / j j \ k ^ k k ' O J J m Y m
= a,a. 2 -2fi^s + 2 o(—7 y) , k J i^i 9i(n~) i^^ > i ( " i ) '
(j / k; j , k=1 ,
a n d \, a, t,
9) Ve ^ Sx) ^k«k^k ^k«k^k Z 1
= e + e 2 —7 r 1=1 ^ i^" i -2 2„. , !! , 1
^^oi + Vk^iki + Vk^2ki) + . ^ ° ( 7 : T ^ ) ' x=1 ^ 1 ^ x^
(^l. = 0> ± 1 f - . IPjjJ k = 1 , . . . , m),
v/here 9.(n.) / 0, 9 . (n . ) —00 as n. -*ooj 1 = 1,...,m,
are necessary'' and sufficient in order that for each f £ D(X ) n. E -* ^ (X) there holds
m ap,X^'
m m (10) L-,(f;X)-f(x) = 2 — r - T [ y .f + 2 ¥, . . f
N i^l 9i(nj_)'- ox ^^^ I j i x^
95
m m m 2 ? , . . f" + 2 ? ' . v - f " ] + 2 o ( — 7 Y ) , I I - > O O .
. ^ 2 j x x . x . , . . 3k J X X, X.- ' . . c p . l n . K ' ] = 1 J J k , j = 1 - ^ k J x = 1 't'x^ 1^
m + 2 ?
' k7 j F u r t h e r , assui.iing t h a t the f u n c t i o n s >?„ . .
2 j x ( j , i = 1 , 2 , . . . , m ) a r e bounded on a bounded s u b s e t S
~ (2) ^^' of X , f o r e a c h f £ D ( X ) ^ E ^ ^ ^ (S ) , ( 1 0 ) h o l d s
m m' a p , X m' ' ^ ^ m
uniformly in X £ S if and only if (8)-(9) hold
uniformly in X £ S . m
96
CHAPTER 2
METHODS OF BOUITDIMG FUIICTIONS AND THE W-PUNCTIONS
Taking into account the unboundedness of given
test functi-ns, in this chapter, v/e extend the
applicability of several earlier theorems ([14],[48],
[50],[56]) on the approximation of unbounded functions
by linear positive operators, based on the methods
of bounding functions and the V/-functions. In accordance
with the common practice, the results are formulated in 2
terms of the test functions 1, t, t ,... . However, it
would be obvious from the proofs that using the same
ideas one can reformulate the results in terms of other
unbounded test functijns as v/ell. All the functions
considered in this chapter are real or complex VEilued.
2.1 Method of bounding functions.
2.1.1 Basic convergence
Let X R and let Q(t) be a non-negative and
unbounded function defined on X. Such a function Q(t) is
called a bounding function. Let I be an index set and
let Q(I,X) = JQ , a £ Ij be a collection of bounding
functions Q defined on X. Let D„(Q,I) denote the class a A^ '
of all functions f defined on X to each of which there
exist an a £ I and two positive numbers A and B such
that |f(t)l < A+BQ (t) for all t £ X, By D°(Q,I) we
denote the class of all functions f £ D (Q,I) to each of
which there corresponds an « £ I such that given an
arbitrary e > 0 there exists a positive number C such
that for all t £ X, |f(t)/Q (t)| < e whenever |f(t)| > C.
Thus D (Q,I) consists of functions having an unboundedne^
97
of a lov/er order than that of some element of Q ( I , X ) .
Similarly D (Q,I) consists of functions v/hose A
unboundedness is at the most of the same order as that
of some member of Q ( I , X ) . Clearly D°(Q,I) C^L^(Q,I)
and also Q(I,X) D (fi,l).
Let I'Y(X) denote the class of all functions defined
on X v/hich with an extension on R are continuous at the
point X £ R. The class of all functions defined on X
which with an extension on R are continuous at each
point of a set S SiR is denoted by L (s),
The follov/ing theorem gives the basic convergence
result for a class of linear operators ultimately
positive on a set of points.
THEOREI'I 1 Let (L , n £ Uj be a class of linear operators
defined on a common domain D(X) (X — R ) of functions
into a domain D ( X ) ( X ^ R ) of functions and ultimately
positive on a set X ^ X , Let Q ( I , X ) . ^ D ( X ) be a 2
collection of bounding functions and let 1,t,t £
D(X) r\ D ( Q , I ) , Let X £ X and assume that for each a £ I A
(I) sup L (Q ;x) < oo ,
n £ U "
Then, in order that for each f £ D(X) r D°(Q,I) \(^)
v/e have
(2) lim L^(f;x) = f(x), n -' oo
it is necessary and suffioient that for 1=0,1,2
(3) lira L^(t ;x) = X . n — oo
Further, if S X is a compact set and (l) holds
uniformly in x £ S for each a £ I, then in order that
for each f £ D(X) r-. J^'^{Q,1) r \ i s ) > (2) holds
uniformly in x £ S, it is necessary and sufficient that
98
for 1=0,1,2, (3) holds uniforraly in X £ S.
Moreover, if Q ( I , X ) S D ( X ) , (X £ X), then in order
that for each f £ L(X) rs D (Q,I) r^ D (x), (2) be satis
fied, it is necessary and sufficient that
(4) lim L (Q ;X) = Q (x) ^ ' n^ a a '
n — 00
for each a £ I and that (3) holds for 1=0,1,2. Also
if Q ( I , X ) S D ( S ) , ( S ^ X is a compact set), then in A
order that for each f £ L(X) r^ D (Q,I) rs D (S), (2)
holds uniformly in x £ S, it is necessary and sufficient
that (5) for 1=0,1,2 and (4) for each a £ I hold
uniformly in x £ S.
Proof. If f £ D(X) D°(Q,I) r Djr(x), it is easily
shown that there exists an a £ I such that given an
arbitrary e > 0 we can find a positive constant A such
that for all t £ X (5) |f(t)-f(x)| < e + A(t-x)^ + cQ|;t).
Using (1), (3) and (5), rest of the proof of the first
assertion of the theorem follov/s along the lines similar
to the proof of theorem 1.1.2.1,
To prove the second assertion, if f £ D(X) r\ D (Q,I) A
r D (x) V/e can show that there exists an a £ I such that
given an arbitrary c > 0 we can find positive constants
A,B such that for all t £ X (6) |f(t)-f(x)| < e + A(t-x)^ + B ( Q (t)-Q (x)),
Using (3), (4) and (6), rest of the proof 16 easily
completed on the lines similar to the proof of theorem
1,1.2.1.
2,1,2 Asymptotic estimates
Eisenberg and V/ood [14] stated the follov/ing
99
theorem referring to the preprint of Muller and V/ "• k
[50],
THEOREM I Let JL j denote a sequence of linear
operators which are positive on (-00,00) and have a
common domain D of functions defined on (-00,00),
Let 1,t,t £ L and f £ C(-oo , 00 ) L, Let
-00 < a < x < b < c n a n d L ( l ; x ) = 1 , L e t T ( t ) = ( t - x ) ^
and u (x) = [L (^ ;x)]^ . If there exists a number p > 1 n^ ' '- n X '-I ^
and a positive increasing (with |t|) function Q(t)
such that Q-P £ D and f(t) = 0(Q(t)) (|t| -' 00), then
for n=1,2,...
(1) |f(x)-L^(f;x)I s 2a)(f,u^) + ra;^|f(x)| u^
+ C(L (QP;x))^/Pm-2/P' u^/?' , \ -Q\ » / / X n '
where l/p + l/p' = 1, u(f,6) = maxj |f(u)-f(v)| :
|u-v| g 6 and u,v £ [a,b]j, C is a constant depending
only on f, and m = min{ | a-x | , |b-x|j.
Taking into account the unboundedness of the 2
function t we obtain the follov/ing improvement on
theorem I.
Theorem 1 Let jL j be a sequence of linear operators
which are positive on (-00,00) and have a common domain
D of functions defined on (-00 ,00), Let 1,t,t £ D
a n d f £ C(-oo ,co ) ^ D, L e t - o o < a < x < b < o o a n d
L ( l ; x ) = 1 . L e t ¥ ( t ) = ( t - x ) ^ a n d u ( x ) = [L {^ ; x ) ]^. j-^\ > / x^ ' ^ ^ n^ ' L n^ X ' ->
If there exists a number p > 1 and a positive increasing
(viith |t|) function Q(t) such that Q- £ D and f(t) =
0(Q(t)|t|^/P' + t^)(|tl -00) where l/p + l/p' = 1, then
for n=1,2,...
(2) lf(x)-L^(f;x)| g 2(o(f,u^)+m-V(|f(x)|+C^)
- Cp(L ( Q P ; X ) ) V P ,,-2/P' 2/P' ,
100
where u(f,6) = max j|f(n)-f(v)l : |u-vl § b ,
u,v £ (a,b)j, m = minj |a-x|,|b-x|j, C and Cp are
constants depending only on f such that C =0 if
|t|^/P = 0(Q(t)) (|t| - «=) and C2 = 0 if Q(t ) = 0( 111 /^)
(|t| - oo).
Proof Define the function \ by
(5) x^ 0 , t £ (a,b)
1 , o ther \ / i se .
It is clear that there exist constants C. and Cp
satisfying the conditions of the theorem and such that
(4) |f(t)| ,^ s C^(i^)2 , Cp Q(t) \ ^ '/'' X X
for all t £ (-00,00) and x £ (a,b). Also a usual
manipulation with the modulus of continuity gives
(5) (1-x^) |f(t)-f(x)| g { 1 + (i^)2j^(r,u^),
n
for all t £ (-00,00) and x £ (a,b). Finally, since
(6) Xjf(x)| § |f(x)| (^)2 , X
for all t £ (-0°, <») and x £(a,b) , the proof of (2)
follows by combining (4)-(6) and an application of
HSlder's inequality,
All the remarks of Eisenberg and Wood [14] made in
connection with theorem I are also valid for theorem 1.
For the sake of completeness with appropriate
modifications we reproduce them here,
Remark 1 If sup jL (Q-P;X) : n=1,2,...j < 00 , then
(7) |f(x)-L^(f;x)| g 2a)(f,u^)+m;;V(|f(x)|+C.,)
^ 2/p' -2/p' + Mu ' - m ' ^ ,
n X
101
where M is a constant depending on x but otherwise
satisfying the properties of Cp in the statement of
theorem 1.
Remark 2 If in addition L reproduces all linear n -
functions, i.e. L (t;x) = x, then u may be replaced P n 2 i n
by t = (L (t ;x)-x )^. Also in this case the modulus " n ^ n^ ' '
of contituity co*(f,6) (see [14] and the reference cited
therein) which is zero if f is linear, may be substituted
for u(f,6). The modulus of continuity u*(f,6) is
defined by w*(f,6) = inf co(f (x)-cx; 6 ). c
Remark 3 If on (a,b), f £ Lip a , we may replace u(f,u ) with Tu". ^ ' n n
Remark 4 An analogous result holds when JL j is defined
on a common domain D(X) where X is not necessarily the
whole of R.
Remark 5 The factor |t| and the term t in the
order 0(Q(t)|t| '-^ +t ) could be included in the 0-
symbol because of the unboundedness of the test function 2
t . In a similar manner using other test functions and
bounding functi jns Q(t) their unboundedness (not
necessarily when |t| -* 00) can also be taken into account.
The criteria for using other test functions is that the
basic convergence result for them implies the basic
convergence result at least f.-r all bounded and 2
continuous functions. Thus the test functions 1,t,t
may be replaced by a unisolvent system or a Tchebychev
system of an order two or more.
Remark 6 The multiplier enlargement may be employed
to obtain analogous results on (- 00, 00) and [0,oo ) in
case the opert'tors L are positive only on a finite
interval. In particular we have
THEOREM 2 Let jL j be s sequence of linear operators,
102
defined on a common domain D of functions defined on
[ 0, oo), which are positive on (0,c), 0 < c < oo and let
{a j be a sequence of positive numbers strictly
increasing to <» . Let 0 g a < x < b < oo, f £ c[0, oo)
and 1,t,t^, f(a t) (n=1,2,...) £ D. Let L (l;x) = 1, If
there exist a number p > 1 and a positive increasing
(with t) function Q(t) such that for n=1,2,...,
Q-P(a t) £ D and f(t) = 0(Q(t) t^/^'+t^)(t - oo) where
l/p + 1/P' = 1> then for n g N(x) v/here N(x) is a
suitably large positive number,
1/p
(8) |f(x)-L^(f(a^t); a~^x) | g 2a)(f,p^)
+ ni-2 P^ (|f(x)|+C^) + C2[L^QP(a^t);a;^x)]
-2/p' 2/p' . m ' ^ P , X n '
where 1 (t) = (a t-x)^, p (x) = [L (? ;a~^x)]'^ , nx^ n ' n^ ' ^ n^ nx n '-^ '
0. and C„ are constants depending only on f such that 9 /
0 =0 if t '^ = o(Q(t)) (t - oo) and C =0 if O I -r\
Q(t) = 0(t ) (t -* oo) and co and ra are as in theorem 1,
Theorem 2 is an improvement over the follov/ing
theorem of Eisenberg and V/ood [l4]«
THE'lHEM II Let JL } be a sequence of linear operators
which are positive on [0,c] , 0 < c < oo , with common
domain D, and let ja j be a sequence of positive numbers
strictly increasing to oo. Let O g a < x < b < o o ,
f £ D o C[0, ex.), and 1,t,t^ £ D. Suppose L (f(a t); a-1x) involves only values of f(a t) for 0 g « t < oo and n ' '' ^ n ' n
L (l;x) = 1. If there exist a number p > 1 and a positive
increasing function Q such that Q £ L and f(t)=0(Q(t))
(t -^ oo), then, for n g N(x),
(9) |f(x)-L^(f(a^t); a;^x)| g 2a)(f,p^)
103
+ m - 2 l f ( x ) | p 2 + C j L {Q\a t ) ; a - \ ) ] ' ^ V ^ / P ' p ^ / P ' , x ' ' n I ' - n ^ ^ n ^ n ^ •' x n
where l / p + l / p ' = 1 , '^^^.(t) = ( a ^ t - x ) S p ^ ( x ) =
[ L ( T ;a x ) ] ^ and C. i s a c o n s t a n t depending on ly on f. ' - n ^ n x n ' - ' 1 •" ° ''
2 . 1 . 3 As.ymptotic f^.rmulae
I n our n e x t theorem v/e improve t h e follovi/ing r e s u l t
of E i s e n b e r g and \/ood [ l 4 ] .
THEOREM I Let [ L j be a setiuence of l i n e a r o p e r a t o r s
which a r e p o s i t i v e on ( - oo, oo), have a common domain p
L, and s a t i s f y L ( l ; x ) E 1. Let 1 , t , t , . . . £ L, Let f,
f £ L and be d e f i n e d on ( - oo, oo) vjlth d e r i v a t i v e s upto
and i n c l u d i n g t h e 2m-th o r d e r a t t h e p o i n t x £ ( - 0 0 , 0 0 ) .
Suppose numbers p . , p > 1 and p o s i t i v e i n c r e a s i n g
( w i t h | t | ) f u n c t i o n s Q , Q e x i s t such t h a t Q.,Qp £ D
and f ( t ) = 0 ( Q ^ ( t ) ) , ? ( t ) = 0 ( Q 2 ( t ) ) ( | t | -> ~ ) , I f
(1) sup i L ^ ( Q . i ; x ) : n = 1 , 2 , . . . j < c« , 1=1,2 ;
(2 ) sup{L ( ( t - x ) ' "^ ;x) : n = 1 , 2 , . . . j < 00 , p=max(p ,p );
[L ( ( t - x ) 2 - + 2 j . ^ ) ^ l / p .
(3) l im -^ — = 0 n - 00 L ^ ( ( t - x ) ;x )
f o r a t l e a s t one v a l u e of j ( j = 1 , 2 , . . . ) , where
1/P+1/P'=1> then
2m-1 /, \ , L ( f ; x ) - f ( x ) - 2 ( f ^ ^ ^ ( x ) / k l ) L ^ ( ( t - x ) ^ ; x )
/ , N , . k=1 (4 ) 1 , ^ 2m-1 .^s —
" ^ ~ L ^ ( ^ ; x ) - Y ( n ) - 2 ( f ^ ^ ^ ( x ) / k ! ) L ^ ( ( t - x ) ^ ; x ) k=1
= f^^ '^^(x)
Note There is a slight error in the statement of above
theorem and its proof given in [14]. In order to render
104
the result correct l/p' in (3) must be replaced by
min(l/p.|,l/p') where l/p^+l/p^ = 1, 1 = 1,2 .
With this change, the correct proof is a minor modifi
cation of the given one.
THEOREM 1 Let {L j be a sequence of linear operators
ultimately positive at a point x £ (- oo, oo) and defined
on a comraon domain D ( - OO, OO). Let m be a positive
integer and j a positive number and let 1,t,t ,...,t ,
|t-x| ™' '' % II(- ~, °°). let L (l;x) = 1, n=1,2,..., and
let ¥,f £ D and possess derivatives upto and including
the 2m-th order at the point x. Let Q,(t), Qp(t) be
positive functions increasing with |t| and let numbers p.
P-iiPp > 1 exist such that Q.- £ L, 1 = 1,2, and let ^2m+2j 2ra+2j
(5) f(t),g(t) = 0[Q^(t)|t| P-1 +Q2(t)|t| P^ +|t|2 -'2j]
as |t| - OO where l/p. + l/p? = 1, 1 = 1,2 . If
(6) n;;: L^((t-x)2'";x) < c« , n — oo
P-i (7) lira L^(fi^^;x) < ^ , 1=1,2,
n -• oo
and
[L^(|t-x| 2-2j^^)^l/p.
(8) lim - ^ ^ = 0 n - OO L^((t-x) ;x)
where l/p + l/p' = 1, p=min (p..,Pp), then
L^(f;x)-f(x)-T\f(^)(x)/k!)L ((t-x)^;x) k£!
( 2m-1 /, N ,
""^'^ L (f;x)-Y(x)- 2 (f^^\x)/k!)L^((t-x)^;x) k=1 "
f(^")(x) - ^(2m)^^)
(9) lim
105
if ?^2ni)(^) Q
Remark 1 Since (t-x) ™ < 1 + |t-x|™'*', v > 1, the
condition (6) is much less severe than the condition (2)
in theorem I. Also (5) extends the applicability of (9)
to a larger class of functions than given by theorem I.
Remark 2 It is interesting (though in an entirely
different direction) to note that the statement of
theorem I remains true, as it is, if v/e replace (2) by
the follov/ing one
(10) lim L (t-x)^'";x) = oo , n — CO
However, then the unboundedness of Q and Q„ is of
little use. A similar assertion holds for theorem 1
as well, if v/e replace p by max (p,.,Pp) and replace
(6) by (10).
Remark 3 It is possible to replace {Q-,Qpj by a set
{ Q. , i £ Ij where I may be vacuous or may consist of
an arbitrary finite number of elements. Then in place
of (5) and (7), respectively, v/e must have
2m+2.i
(11) f(t),f(t) = 0( 2 Q.(t)|t| ^i +|t|2'"+2j) (|t|-.oo) i £ I
and P-i
(12) lim L (Q. ;x) < 00 , i £ I, n — 00
P i w h e r e p=min { p . , i £ l j , p . > 1 , i £ I a n d Q. £ D(-OO,OO).
If I is empty, theorem 1 reduces to a generalization of
a result of Mamedov [36].
Remark 4 Remarks 2.7-2.9, [14], p. 269, made by
Eisenberg and V/ood at the end of the proof of theorem I
remain valid for theorem 1 as well, V/e also refer to a
remark (Rathore [54], P. 145) made in connection with
106
theorem I, in the light of local unsaturation on zero
orders, generalizing the above limit ratio phenomena of
(4).
Proof of theorem 1 Borrowing the notations used in the
proof of theorem I ([14]), we have, given an arbitrary
£ > 0 a 6 > 0 such that
-e(t-x)2°' g |)(t)-Kx)-A(t-x)2'° g e(t-x)^"'
for a l l t s a t i s fy ing | t - x | < 6. For | t - x | g 6 we have
a constant C(6) such that 2m+2.1
|^(t)-(^(x)| g C(6) { 2 Q.(t)|t-x| ^^ +|t-x|2'"+2jj,
1=1
Thus
|4(t)-(l)(x)-A(t-x)2'"| 2m+2,i
g C(6){ 2 Q.(t)|t-x| ^^ + |t-x|2'"+2^| i = 1
^ U ^2m , J A I 14- |2m+2j + E(t-x) + -hrf It-xI 6 ^
Rest of the proof can be completed by an application of
HSlder's inequality and using (6)-(8).
Next we come to the follov/ing two theorems of
Muller [48], in which no direct assumption is made on
the order of the growth of f but only a local assumption
on the L -images of |f| is required. In the rest of this
section we use Mtiller's notations in which Q denotes a
non-void (bounded or unbounded) domain of the real line,
D(S2) the linear space of all complex-valued functions f
defined on Q and Dp (Q;X) the linear sub.space of those
functions f £ D(Q) which possess derivatives upto and
including the 2p-th order (p=1,2,...) at a fixed point
X £ Q.
107
THEOREM II Let f € D2(Q;X) and let (L^j (n=1,2,...) be
a sequence of linear positive operators mapping D(Q)
into D(Q) which satisfies the conditions
L (l;x) = 1 + O(-TU-) n ' ' 9(n)' -,(x) ^
(13) L (t;x) = X + / \ + o(—7—v) ^ ' n^ ' ^ 9(n) ^9(n)^
2 2 2 ^^' 1 L (t ;x) = X + —7—r- + o(—7—r) n^ ' 9(n) ^9(n)'
and
(14) lim L (|f|-' ;x) < 00 (v > 1 a real constant). n ^ 00
If there exists a natural number m such that
(15) x[''"^^^(x) = Lj(t-x)2-+2.^^ __ o ( _ ^ - ^ ) ,
[9(n)]^
l/r+l/r' = 1, for n — 00, then the local degree of
approximation to f by the sequence {L f j(n=1,2,...)
is given by the asymptotic formula
2T.,(x)f'(x) + {?2(x)-2xY^(x)jf"(x) (16) L^(f;x)-f(x)
n ~^U^
(rTTTT) . + 01 / s >(n)'
THEOREM III Let f £ l'2m+2 ' '' "^ •'"®* I ^ 5 "''' 2'* * *
be a sequence of linear positive operators mapping D(Q)
into D(Q) and satisfying the condition (14) with a real
constant r > 1. If there exists a natural number j such
that
T [2m+2j+2], ^,1/r' n ^^'^ „ 1 1 •17) lim ^ — 7 7 ^ = 0 , - + -L = 1
• ' 2m+2 \ i \ > -£, -^i » n - ' T^ -i(x) n
then
.,,. ,. V^2m+1'") f(2" 2)(x) ( ) 1" [2m+2], s = 7 2 ^ ^
108
where
2m+1 ^(k)/ N
(19) R2m+l(^) - ' ^ ' ^ - ' , ^ ^ ( t - x ) ^
k=0
An improvement over theorem II is as follows.
THEOREM 2 In the statement of theorem II, (14) can be
replaced by the more general condition 2m+2
(20) li^ L^(|f(t)/(l + jt| ' )|'';x)<oo, n -' 00
Also, if f is bounded on all bounded subsets of Q, then
in (15) the o-term can be replaced by a corresponding
O-term, the other hypotheses of theorem II reraaing
unchanged.
In a similar way, theorem III can be improved to
give
TIIEOREM 3 In the statement of theorem III, (14) can be
replaced by the more general condition
2m+2j+2
(21) Ti^ L^(|f(t) /(l+|t| ^' )|^;x)<o^. n - 00
Also, if f is> bounded on all bounded subsets of Q, then (17) can be replaced by
/^-^ r-r— r2m+2j+2]/ N,l/r'/ [2m+2]/ v (22) lim T- ^ J(x)j / T^ •J(x) < CO,
n -* cc
provided
(23) lim T^2'"+2^(x) = 0, n -• 00
the other hypotheses of theorem III remaining unchanged.
Remark 1 It is obvious that (14) implies both (20)
and (21) but that the converse is not true in general.
Remark 2 Replacing (t-x) by |t-x| , m in
theorems II and 2 can be taken to be an arbitrary
109
positive number not necessarily an integer. Analogous
result holds for j occuring in theorems III and 3.
Proof of theorem 2 Let us start from the relation
,2 (24) f(t)-f(x)-f'(x)(t-x)-f"(x)
(t-x)-2
2m+2
= g^(t)(t-x)2+h^(t) - i - ^ |t-x| r-
1+|t| ^'
/ 4. 14. I 2m+2 + P^(t) |t-x|
It is easy to check that the unknown functions g (t),
h (t) and p (t) can be chosen so that given an
arbitrary e > 0 there exist positive constants A(e) and
B(e) so that |g^(t)| < e,|h^(t)| < A(e) and |p^(t)|< B(e)
hold for all t £ Q . Rest of the proof of the first
assertion consists in an application of HSlder's
inequality and some routine manipulations.
The proof of the second assertion is an easy
modification of the proof of the first assertion and is
therefore omitted.
Proof of theorem 3 Writing
„(2m+2)/ \ o ,o
(25) ^2m+l(^) = ( x ) - (2m+2ST^) (^-)'"^'' 2m+2j+2
|f(t)| It-x 3:' o ,o- o + h (t) ' "p p. , + p (t)|t-x|2'"+2a+2
x^ ' 2m+2j+2 ^x^ ' ^ I '
1 + |t| r'
as in the proof of theorem 2, one easily verifies that
in this case as well, the functions g (t), h (t) and
p (t) can be chosen so as to satisfy similar constraints,
i.e., for an arbitrary e > 0, |g^(t)| < e ,|h^(t)|< A(E )
and |p (t)| < B(G) for all t £ Q, where A(e) and B(e) are
positive constants. Thus using Holder's inequality we 110
have
(20 |Ln( 2m+l( )'-) - T^^SyH \(i^-^f'-'';-)\
g eL^((t-x)2"'+2.x) + A(e).
[ nU 'i:!j,.2i^^-)]^^^tvi^-i^^^^w^' 1+|t| ^'
+ B(e) L^ (|t-x|2^+2j+2.^)^
Dividing the inequality (26) by T"- •' and taking
limit as n -• oo, (18) follows as a consequence of the
arbitrariness of e >0. The proof of the second assertion
is an easy modification of above proof and is therefore
omitted.
2.2 Combinin,-? the techniques of bounding functions
and the W-functions
In this section we generalize and improve upon
several results of Schmid [56], based on the concept
of a V/-function, by recasting them in terms of bounding
functions and taking into account the unboundedness of
given test functions.
2.2.1 Asymptotic formulae
In the sequel the functions denoted by Q(t), Q (t)
(n £ U) etc. are taken to be arbitrary non-negative and
unbounded bounding functions defined on a subset X of R.
In particular, the follovring may be convenient choices
for the functions Q (t)(n £ U):
(a) Q^(t) = ej h( |f(t) | ) , where h is a fixed V/-
function, f is the function to be approximated and
e -• 0 as n -» oo. n
(h) Q (t) =e Q (t), where Q(t) is a fixed bounding
111
function and e -* 0 as n -* oo, n
(c) Q (t) = Q(t), where Q(t) is a fixed bounding
function (depending on the points at which the functions
are to be approximated) such that it has zeros of
suitably high orders at the points where the approxi
mation is desired.
THEOREM 1 Let {L , n £ UJ be a class of linear
operators defined on a comraon domain D ( X ) ( X S R ) of
functions into a domain D(X)(X £r R) of functions and
ultimately positive on a set X X. Let{Q (t),n £ Uj Sr
D(X) be a set of bounding functions. Let m be a positive
number and let 1,t,t ,|t-x| £ D(X) for all x £ X. Let
X £ 5! and f £ D(X) such that with an extension on R, f
is tv/ice differentiable at the point x. If there hold
the conditions
^o(x) ^ L (l;x) = 1 + —7—Y + o( V v) n^ ' 9(n) ^<5)\n)'
^ (x) (1) L (t;x) = X + "*/ V + o(—r-r) ^ ' n^ ' 9(n) 9(n)'
2 2 2 1 L (t ;x) = X + —7—V + o(—7—r) , n^ 9(n) ^9(n)' '
(2) L (Q ;X) = © ( - T W ) , and ^ ' n^ n ^9(n)''
(3) C^(2m+2,a)42-2]^^) __ ^_^^ ^
as n -• OO, where 9 ( n ) / 0, 9 ( n ) -» oo as n -• oo, Y. ( x ) ,
1 = 0 , 1 , 2 , a r e f u n c t i o n s i n x , a i s a n o n - n e g a t i v e r e a l
c o n s t a n t ,
| f ( t ) | - Q ^ ( t ) (4) C^(2m+2,a) = sup { _ — _ _ ^ j , and
t £ X 1+ aft I
(5) 42ra+2]^^) =L^(|t-x|2--2.x) ,
112
then there holds the formula
(6) L^(f;x)-f(x) = ^ [f(x)^^(x)
+ f ( x ) {Y^(x)-xY^(x)) + ^ ^ { f2(x) - 2x^P^(x) + xh^{x)]] + o ( ^ ) , n ^ oo .
Proof. For an arbitrary 6 >0 define for all t £ R
0 , |t-x| < 6
.,(t) = \
1 , |t-x| g 6 .
Then we have for all t £ X
|f(t)|,.^(t) g C^(2m+2,g)(l+a|t|2°^-*-2)^^(t) + Q^(t).
Clearly there exists a constant A > 0, depending on x,
such that for all t £ R /., |,|2m+2N /,^ ^ ,1, |2m+2 (1+a|t I )(ij (t) g A I t-x I
Hence for all t £ X
|f(t)|„^(t) g A C^(2m+2,a)|t-x|2^+2 ^ ^^^^
Rest of the proof is evident from (l)-(3) along the
lines of the proof of theorem 1.1.3.1.
Remark 1 Putting Q (t) = e h([f(t)|), a=0 and taking
m to be a positive integer, Satz 1.1, p.16, [56] of
Schmid follows.
Remark 2 Theorem 1.1.3.1 can be obtained as a corollary
of theorem 1 .
Remark 3 Por continuously twice differentiable
functions, uniformity of (6) can easily be incorporated
in the statement of theorem 1,
THEOREM 2 Let {L , n £ U} be a class of linear operators
defined on a common domain D ( X ) ( X ^ R ) of functions into
a domain D(J(!)(X^R)of functions and ultimately positive
on a set l ^ t . Let {Q (t), n £ U}^D(x) be a set of
bounding functions, m a positive number, p a positive
113
integer and let 1,t,t ,...,t •P,|t-x| •^ £ D(X) for all
X £ X. Let X £ X and f £ D(X) such that with an
extension on R, f is 2p-times differentiable at the
point X. If there hold the conditions
/r.\ -I • L ( Q ;X)
(7) Ixra n^ n ^ n-oo;j2i7( ^ 0 , and • -Ifx)
n
C (2ra+2p,a)42in+2p](^)
(O ^ - [2p], ; = ° ' n-*oo T ' - - ' ( X )
where
42P](x) = L^((t-x)2P;x) ,
^[2-2p]^,^(,,_^,2m+2^^)^ and
|f(t)|-fl^(t) C (2m+2p,a) = sup { 2m+2p ^ '
t £ X 1+a|t| P
where o is a positive number, then there holds
n ^ '
where
2p-1 M(^ ^ (10) R ( t ) = f ( t ) - 2 S r r ^ ( t-x)^.
P~' k=0 ^•
for all t £ X.
Proof. For an arbitrary 6 >0 let n. (t) be defined as
in the proof of theorem 1. We then have for all t £ X
I^-Z^^I^'S^*) = Cj,(2m+2p,a)(l+a|tl2"'+2P)^^(,)^.Q^(t).
As we can find a positive constant A such that for all
t £ R
(l+altl^'^^^p)^^^^^ g^|^_^|2m+2p^
we have for all t £ X
114
(11) |R2p_i(t)|Hj,(t) g A C^(2ra+2p,a)|t-x|2'^+2p ^ ^^^^^^
Having shown this, since given an arbitrary e > 0 we can
find 6 > 0 such that for all t £ X and satisfying
It-x] < b
( 2) l%-i(0-%iy]^^^'"^(-)l <^(t-x)2p, the proof of (9) is obvious from (7), (8), (ll), (l2),
the arbitrariness of e > 0 and the positivity of the
operators.
Remark 1 Putting Q = e h(|f|), a = 0 and taking m
to be a positive integer, Satz 3.1, p. 35> [56] of
Schmid is obtained as a corollary to theorem 2.
In case, v /ith an extension of f on R, only the
left and the right derivatives f^^^^ (x) and f^^^^x) •— +
exist without necessarily being equal, we have
THEOREM 3 Under the remaning hypotheses of theorem 2
there holds ^n(^2p V") f(2P)(x)+f(2p)(^)
(13) lim [2P5" . = 2-(2^l n — oo x'- -'(x) ^ ^ ' n
where Rp _ is given by
2p-1 „(k) (2LI (^_^\^ (14) R* .(t) = f(t) - 2 ^ - T ^ it-x)'
P J Q
r(^P)(.)-f(^'')(,) I,. ,,,.,,2P-,
2 (2p)!
for all t £ X.
Proof. Using the knov/n i d e n t i t y (see p . 42, Schmid [56])
('^) , - i g ^ - M a p ) , -
115
The proof of theorem 3 is essentially the same as that
of theorem 2,
Remark 1 Putting Q =e h(|f|), o=0 and taking ra to be
a positive integer, Satz 3.2, p.415 [56] of Schmid
follov/s as a corollary to theorem 3»
In case v rhen with an extension of f on R, the right
and the left derivatives f (x) and f (x)
exist v/e have the follov/ing result.
THEOHEIi 4 Let jL , n £ Uj be a class of linear operators
defined on a common domain D ( X ) ( X £ R) of functions into
a domain D(X)(XS: R ) of functions and ultimately
positive on a set 'X S X. Let JQ (t), n £ Uj D(X) be a
set of bounding functions, mi-a positive number, p a
positive integer and let 1,t,t ,..., t ,|t-x| ,
I _^ I 2m+2p+1 ^ ^ - ^^_^ ^-^-^ X £ J.. Let X £ 'X and f £ D(X)
such that with an extension on R, f is 2p-times
differentiable at the point x and that the derivatives
f(2p+l)^^^ and f[^^^^\x) exist. If there hold the
conditions
(16) lim rn^,<-i = 0, and n^oo xL2P+^J(x)
C (2m+2p+1,a)xt2--^2p+1](^^) (17) lira -^
n -«• 00 ^[2P-1](,) n ^ '
v/here
?[2P-1](,) =L(lt-x|2P-^ n ' ' n~
;x, I
J2m+2p+1]/ > . /u |2m.+2p+1 s , Ti; -"(x) = 14 ( t-x ;x), and
n
r i
Jf( t ) | -Q^(t) (2m+2p+1,a; = sup
-^ ^ '"^ t £ X S + a | t | 2 - 2 p . 1
where a i s a pos i t i ve number, then v.-e have
116
L^(R ;x ) f ( 2 P - ^ ^ ) ( x ) - f ( 2 P - ^ ) ( x )
(^Q) 1^^ . [ 2 p + l ] , . = 2 . ( 2 p + l ) ! ^ n -» oo x"- (x) ^ ' n ^ '
where Rp i s g iven by
2P 4.(k) ( 1 9 ) R p j t ) = f ( t ) - 2 ^ . i^^ ( t - x ) ^
• k=0 „ ( 2 p + l ) / X .o(2p+l) / \ o ., f;; ^ H x ) + f ^ ^(x) (t-oc}fP+]_
2 (2p+1) !
f o r a l l t e X.
P r o o f . To p rove theorem 4 we use t h e knovm r e s u l t
( p . 44, Schmid [ 5 6 ] )
H2p( t ) f i 2 P + ^ ^ ( x ) - f p P - ^ ) ( x ) ( 2 0 ) l i m — ' ^—T;—7 = ;;—r;;—rv; ^ t ^ x ( t - x ) 2 P + 1 2 . ( 2 p + l ) !
and proceed in a vay analogous to that of the proof of
theorem 2.
Remark 1 Putting Q =e h(|f|), o=0 and choosing m
sttch that 2m+1 is an even positive integer, Satz 3.3,
p. 42, [56], of Schmid follows as a corollary to theorem
4.
2.2.2 Asymptotic estimates 2
Utilizing the unboundedness of the function t
(as |t| -•00) we now obtain upper bounds of the error
in the approximation of continuous and continuously
differentiable functions by a class of linear positive
operators.
THEOREI l 1 Let {L , n £ U} be a class of linear
operators defined on a common domain D ( X ) ( X ^ R ) of
functions into a domain D(x) of functions and positive
on a set Sf SiX. Let I = (a,b) be an open interval
contained in K and let the functions 1,t,t ,f(t), Q(t)
117
£ D(X) where f(t) is bounded and continuous on I and
Q(t) is a bounding function. For x £ I let
L (l;x) = 1 + a (x) , n^ no^ ' '
(1) L^(t;x) = X + ot .](x),
L^(t ;x) = X + g^2^^''
where a .(x) — 0 as n-oo, 1 = 0,1,2; x £ I. Then for ni '
arbitrary e,6 > 0, for each x £ I v/e have
(2) |L^(f;x) - f(x)| g M(6,n,x) (0j(f;6)
+ CL^(Q;X) + |f(x)|(m;2xt2](^) + |a^^(x)|)
+ m-2 C(2,a;e)T[^^(x) [l + a(lx| +mj^]
where
M(6,n,x) = l+|a^^(x)|+min{6-^[x[^^(x)]*,6-2xt2](^)|^
m^ = min {|a-x|,|b-x|j,
' i -' ) = L^((t-x)^x) = g^2(^)-2xan1^^)+^^«no(^)'
C(2,a;E) = sup {Mtlb_eQ(tij^ ^^^^^ ^ g Q
t £ X l+gt
and (i)-p(f;6) is the local modulus of continuity of f on I
defined by
u,^(f;6) = sup(lf(t^)-f(t2)|: t^,t2 £ I, jt^-tpl g 6}.
Proof Let Xj denote the characteristic function of the
interval I and put i-j- = 1-\-j.. Por all t £ X, x £ I v/e
have then
(3) |f(t)-f(x)| g (Oj(f;|t-x|)\^+(|f(t)| + |f(x)|)nj.
Now
(4) a)j(f; lt-x|)\j g [l+minjs"^ |t-x|,6~^(t-x)^}]a)j(f;6),
118
(5) |f(t)|jij g (ij(l+at2) C(2,a;e) + £fi(t)
g m~2(t-x)2{l+a(|x|+m^)^}c(2,a;E)+efl(t),
and
(6) |f(x)| nj Sm-2(t-x)2|f(x)|.
Combining (3)-(6) by the linearity and positivity
of L , n £ U, the inequality (2) is immediate.
Remark 1 Taking g=0, Q(t) = h(|f(t)|) where h is a
W-function Satz 4.1, p.46, [56], of Schmid follows. It
is clear that if f(t), Q(t) have unboundedness as
|t| -» oo then a positive choice of a will keep G(2,a;E)
relatively smaller than in the case of a=0. Thus in
such c^ses (2) vi/ill result in a sharper estimate than
given by the result of Schmid.
Remark 2 In his result Schmid takes I to be the closed
interval [a,b]. However, then the function m~ is
undefined for x=a,b. For this reason vre take I to be the
open interval (a,b).
If f is differentiable on (a,b) and if f is
continuous and bounded there, then, with (o-p(f';6) to be
the modulus of continuity of f on (a,b) v/e have
THEOREM 2 Under the assumptions and notations of
theorem 1, for arbitrary e,& > 0, a = 0 and x £ I, in
this case we have
(7) |Ljf;x)-f(x)| g {[ ^[^^(x)]Kt-\l^\x)]o^^if';t)
+ eL^(Q;x) + |f (x) | (m;^^^]^^) +|ano^^)|)
+ m;2 C(2,a;s)4^](x)[l+a(|x|+m^)2]
+ |f'(x)| {ml\[^'^M + \a^^(x)-xa^^(x)\}.
The proof of theorem 2 is an obvious adaptation of
the proof of theorem 1 and so is omitted.
119
Taking g=0, Q(t) = h(|f(t)|), where h is a W-function,
we obtain Satz 4.2, p. 54, [56] of Schmid. The choice
g > 0 has the same advantages as given in remark 1
following theorem 1.
In connection with Satz 5«1> P. 57 and Satz 5.2,
p. 60, [56] of Schmid (concerning error estimates for
multiplier enlargement) we remark that they do not
require a separate proof. A.t a first reflection it is
clear that these are simply re-statements of the
earlier Satz 4.1 and 4.2 respectively, if we replace
JL { by the operators {L }. In this light v/e omit error n
estimates for the multiplier enlargement technique; they
can easily be derived from theorems 1-2.
120
CHAPTER 3
APPROXIMATION OP UNBOUNDED FUNCTIONS BY OPERATORS
OF SUCTIATION TYPE
In this chapter we indicate an application of
certain estimates closely connected with the local
unsaturation [54] of linear positive operators on
positive zero orders, mentioned in the introduction of
the thesis, to determine the approximation of certain
unbounded functions by operators of summation type.
3.1 A general outline
In the sequel we limit our treatment to the
Bernstein polyn'jmials and obtain certain classes of
unbounded functions which can be approximated by the
sequence of these polynomials at all points of contituity
of the functions. Here also we limit ourselves to proving
only the basic convergence result for these classes and
do not go into obvious details by means of which various
results on asymptotic estimates, asymptotic formulae,
simultaneous approximation of derivatives of functions
by the derivatives of the polynomials, linear
combinations and linear com.binations of iterates etc.
can be extended to the same classes of unbounded
functijns.
The tecnnique used serves as a prototype fir all
sequences of linear positive operators of summation type
for which asympt -tie estimates of the local unsaturation
order (see the Intr duction) or even those of certain
moments or of approximation of certain zero orders are
available. Por the Szasz-Mirakyan-Hille operators
vari us results as v/ell as their proofs are almost word
121
to word the same as those for the Bernstein polynomials
(with the difference of the intervals [0,l] and [ 0, oo)
on which they are respectively defined). As an
application of a mere knowledge of the asymptotic
behaviour of various moments, we remark, for instance,
that the Laguerre-Bernstein series of Cheney and Sherma
([9],[54]) can approximate functions having an
unboundedness of order t (b > O) as t -* 0 (for special
subsequences a similar result nolds for functions
unbounded in the neighbourhood of an interior point in
[0,1]).
3,2. The Bernstein polynomials
Let f be a function defined on the interval [0,l].
The Bernstein polynomials B (x), n=1,2,..., of the
function f are given by
(1) B^(x) = 2 p^^(x) f(J), X £ [0,1], v = 0
where
(2) p^^(x) = C^)x''il-x)''-\ (n=1,2,...; v = 0,1 n).
The follov/ing result which is a modification of
Theorem 1.5.3, PP. 18-19, Lorentz [33], gives a basic
estimate v/hich v/e require in the sequel.
THEOREM 1 Let 6 > 0. If for all x £ [0,l] there holds
the inequality
(3) S p^ (x) SA(6) e-*<"''(°',
IS-1 * n=1,2,..., v/liere A(6) and 3(6) are positive functions of
6, independent of n and x, then if 9(n) ^ 0, n-1,2,...,
the sequence [(p(n)/nj , n=1, _,..., is bounded. Also, if
122
0 g X g 1 and 0 g 6 < ^, then there holds the inequality
.2 (4) 2 p^ (x) g 2e"° " , n=1,2,...
F-x| 6 'n '
Proof. First v/e prove the second assertion. Proceeding
as in the proof of theorem 1.5.3, PP. 18-19,[33] , if
|u| g 3/2 and x £ [0,1] , then
U ( 1 -X) / . N -ux ^ . u
xe ^ + (1-x)e g 1 + -7- .
Thus A ( \ def _, u(v-nx) , N 0) (u,x) = = 2 e ^ ^p (x) 'n ' ' „ - nv
v=0
= [xe^^^-^)+(l-x)e-^]''
2 ^ nu g exp 4 •
If
n I 1 ,„ / N r, u v-nx / \ ? (u,x) = 2 e 1^ Ip (x), n^ ' „ m v v = 0
then for 0 g u g 3/2, 2
j (u,x) g i)^(u,x) + (| (-u,x) g 2 exp ~ .
Hence for u,c ^ 0
2 Pj (x) g ^ , exp(u|v-nx|) g c? (u,x) ^
which implies that
2 2 P (x) g - . ( \ M -. o nu"^ nv c ,
exp(u|v-nx|J g 2c exp —— / 6^n
Putting c =- ~|- e a,nd u=26 , the second assertion of
the theorem follov/s.
Por proving tho first assertion of the theorem,
assume that for a subsequence {n } , say, of the
sequence fnjof natural numbers, there holds 123
9(n^) lim
Then for an arbitrary positive 6 n^ n^-B(6)9(n^)
2 P^ ..(x)e g A(6)e
— 0, as n —
uniformly in x £ [0,1] . However
Pn /^)® = "* x(l-x) n* , "#-^ "*
•X-
n„x n, = — {(l-x)ej
-• oo , as n -* oo ,
for each x £ (0,1-e ). This contradiction proves the
first assertion, and the proof of the tneorem is
complete.
The follov/ing theorem generalizes Theorem 1.9.2, p
28, Lorentz [33].
THEOREM 2 Given an arbitrary x > 0, there exists a set
C S[0,1] of measure 1 such that if f is any function
defined on [0,1 ]such that (a)
(5) f(t) = 0 (exp |t-c|"' ' ' '' ), as t - c,
for every c £ C and (b) if jt, j, k=1,2,..., is any
sequence of numbers such that the sequence {f(t, )j is
unbounded then Jt, } has a limit point c £ C, then there
holds the relation
(6) lim B^(x) = f(x) n — oo
where x i s any point of con t inu i ty of f. Moreover, i f f
i s continuous at each x £ [ a , b ] ^ [ 0 , l ] , then (6) holds
uniformly in x £ [ a , b ] .
Hov/ever, i f B(v) i s a pos i t i ve and rionotonic
124
function defined for v > 0 and diverging to infinity as
v-»oo,then ther-e exists a set C * ^ [0,l] of the power of the
ontinuum such that for any non-negative f defined on
[0,1] if there holds
(7) Edt-c*!""") = 0(f(t)), t - a*(t / c,*),
where c* is some point of C*, then the sequence JB (x)}
is unbounded for every x £ (0,l).
To prove theorem 2 v/e require some basic results.
First we prove a general lemma. Por Bernstein
polynomials v/e require the result of this lemma
restricted to the interval [0,l].
LEMl iA 1 Let C be a subset of the extended rea-1 line
R* obtained by adding the points + oo to R and let to
each point c £ C there correspond a positive function
g (t) defined on R such that g (t) is continuous for
t / c and g (t) — oo as t -• c. Let f be a function
defined on R and possessing the following properties
(a) f(t) = 0(g^(t)), as t - c (t / c),
for every c £ C, and
(b) if Jt, } is any sequence of real numbers such that
|f(t, ) I -• oo as k -• oothen J t, } has a limit point c £ C.
Then there exists a finite and bounded subset C of tho
set C such that given any two positive numbers A and 6
there exists a positive number A., say, such that o
|f(t)| < A^
for all t satisfying both |t| g A and |t-c | g b for
each c° £ 0 ° . \ '
Proof Let C* be the subset of C consisting of all finite
c such that the set Jf(t) : t £ (c-6,c+6)j is unbounded
for every positive 6. If C* is finite then with C = C*
125
the assertion of the lemma is satisfied. For if not,
then for some A, 6 > 0 the set Jf(t):|t| g A and
|t-c I g 6 for each c £ C } is unbounded. Let Jf(t, )j
be a sequence of members of this set such that
|f(t^^)| -• oo as k — oo. Then Jt, j has a limit point c £ C.
Clearly c £ C . But, since the set Jt:|t| g A and
|t-c I g b for all c £ C j = A , say, is compact,
also c £ A . A s G ^ A = 0 , this gives a contradiction.
Next let C* be infinite, then C* has a limit point,
say, c* £ R*. If |c*| < oo , then for each sufficiently
small 6 > 0 there exists a positive number A , say, o 0
such that
|f(t)| < A g^^(t), for all t £ [c*-6^,cJ+6J. o o
By continuity of g „(t) for t / c* , it follov/s that C* 0
o on every closed subset of the set [c*-6 ,c*+6]-}c*},
' - O O O ' O '
f(t) is bounded. Also there exists a c*£ C* (c* / c*)
such that c* £ (c*--g-6, c*-t -6). Hence there exists a
closed interval [a,b] contained in one of the intervals
(c*-6 ,c*) or (G*,C*+6) such that c* £ (a,b). But then 0 0 0 0 0 \ r /
f(t) must be unbotinded on [a,b], which is a contradiction. It follov/s that c* coincides with one of the points + oo. o —
Let c* = +00. Then if A is a sufficiently large positive
number, there exists a constant B > 0 such that |f(t)| < B g^^(t), for all t g A.
o
Also it is clear that for some positive number k, there
exists a c*£ C* such that A < c* < kA. Consequently f(t)
must be unbounded on the interval [A,kA]. However,
the continuity of g ^(t) implies other\/ise. This contra-0
d ic t ion shov/s tha t c* / + oo. In a s i m i l a r \/ay we can
126
shov/ that c* / - oo. Hence C cannot be infinite. This o '
completes the pr,of of the lemma.
DEFINITION 1 (p. 28, [33]). Let A(V) be a positive and
monotonic function, defined for v > 0, and such that
A(V) -» 0 as V -• OO . Then a real number c is said to
admit the approximation A ( V ) , if there is an infinity
of positive integers n such that |c-v/n| g A(n) is
fulfilled for some integer v. In case we restrict n to
belong to a fixed set I, say, and if the above property
is satisfied with n £ I, we say that c admits the
approxiraation A(v) through I.
DEFINITION 2 Let A(v) be a function as above. The
auxiliary function A*(t) for t > 0 is defined by
(8) A*(t) = [1+inf {v : 1/A(V) > tj]""".
We also need a result of Koksma [25] (see Lorentz
[33], p. 28).
THEOREM I Alraost all real c do not admit the approxima
tion A(V) = V~ , k > 2. Also, for any A(V) -* 0 for
V -• 00, there is a set of the c of the pov/er of the
continuum which admit the approximation A(v).
Proof of theorem 2 Let 0 < x < T. By Koksma's theorera
alraost all real c do not admit the approximation
A(v) = v~ ' ' o . Let C be the set of all such c £ [0,l].
With this C we prove the first assertion of the theorem.
First we show that there exists a positive integer
m and c.,c ,...,c £ C such that for an arbitrary 6 > 0
we can find a positive number A such that
|f(t) I g A^, for all t £ A-,,
vAere A . j = [ 0 , l ] ~ J t : | t - c . | < 6 f o r some i = 1 , 2 , . . . , r a j .
To each c £ C d e f i n e
127
.expj|t-c|~^2"^'') ) , t £ [0,1] , t / c, (9) g (t) =
M , t = c.
Restricting to the interval [0,1]identify the set C and
the functions g (t), c £ C, defined above v/ith the
corresponding notations in lemma 1. If f satisfies the
conditions of the first assertion of theorem 2, it
follov/s from lemma 1 that there exists a finite subset
C S C such that given an arbitrary 6 > 0 there exists
a positive number A such that f(t)| g A., for all o
t £ [0,l] and satisfying |t-c | g 6 for each c £ C .
Designating the elements of C by c ,c , . .. , c , say,
our assertion follov/s.
Now, v/e can choose a 6 > 0 so small that there
holds |f(t)| g A ^ expj|t-cJ-(2+x)-^j
for all t satisfying |t-c | < 6 , t / c , p=1,2,..,,m,
v/here A , p=1,2,...,m, are suitable positive numbers
independent of 6. Let M = max |f(c )| and put 1 g p g ra P
A = max (A,, M,A.,A„,..., A 1. ' 6 ' ' 1' 2' ' m'
Then v/e have
m (10) |f(t)| g A 2 g^ (t) , t £ [0,1] .
p=1 p
Since c.,c ,,...,c do not admit the approximation 1' /2' N ' m
A(v) = v~^ "^o^ it follows frora (lO) that for all n
sufficiently large v/e have
2-^-0 n -5-—
lf(J)| g ra A e ^^^ , v = 0,1,2 n.
3 By theorem 1, t h e r e f o r e , v/ith b <. -f , we have
128
H Pn,(-)|f(i)l ^ 2 m A e"^'-^-'^' ,
F-x| g 6 'n '
for all X £ [0,l] and for all n sufficiently large.
Hence for an arbitrary 6 > 0
(11) lim 2 p^^(x) f(J) = 0
^^°° |^x| g 6
uniformly in x £ [0,l]. From (ll), the first assertion
of theorem 2 is obvious.
To prove the second assertion of theorem 2, we
utilize the second part of theorem I. It is clear that
if B(V) satisfies the conditions of theorem 2, then
there exists a positive and monotonic function g(u)
defined for u > 0 such that g(u) -» oo as u -•ooand 2
B(g(u)) g e" , u > 0.
Put A(V) = l/g(v), V > 0. Then A(v) - 0 as v - oo and
so by Koksma's theorera there exists a set C* o [0,l],
say, of the j)ower of the continuum each member of which
admits the approximation -'i(v). Without loss of
generality we can assume C* to consist of only irrational
points. With this set C* v/e prove the second assertion
of the theorom.
Let f be a function satisfying the conditions of the
assertion of the theorem and let x £ (0,1). Let c* £ C*
be such that (7) holds. Let [(v »n. ) } be an infinxte
sequence of distinct pairs of integers satisfying
,x c l g A(n„) .
Since B(|t-c*| '') = 0(f(t)) as t - o*, there exists a
129
positive number A such that for all n sufficiently large V V
n„ , n, f(-^)p (x) g A" B(|—^ - c*r^)p (x)
^B(|^-c.|-^
where q i s t h e s m a l l e r of t h e numbers x and 1-x. Nov/, Vn n2
B ( | — ^ -c*\~^) g B ( g ( n ^ ) ) g e * . *
I t fo l lov/s t h a t f (v / n ^ ) p (x) — oo as n^ — oo . n^' * X v n *
Thus the sequence jB (x)} is unbounded for every x £(0,l)
This completes the proof of theorem 2.
THEOREM 3 Let I be an unbounded set of positive
integers denoted by n^, let C be a set of numbers
c £ [0,l] and to each c £ C let there correspond a
function A (v) satisfying the conditions of definition 1
such that c does not admit the approximation A (v)
through I. Let f be a function defined on [0,l] such
that if Jt, j is any sequence of real numbers such that
|f(t, )| -• oo as k -» oo then Jt, } has a lim.it point c £ C
and that for each c £ C and every a > 0
(12) f(t) = 0(expj r-i) , as t - c (t / c).
A*(|t-cr^)
Then f o r each x £ [ 0 , l ] where f i s con t inuous t h e r e holds
(13) lira B^ (x) = f ( x ) . n„ n -» oo •
Further (13) holds uniformly in x £ [ a , b ] ^ [0,l] if f
is continuous at each x £ [a,b] .
Proof. In order to prove theorem 3 it is sufficient to
show that, under the given hypotheses, for an arbitrary
6 > 0
130
lim 2 p (x) f ( ^ = 0 m„v n„^
'n '
uniformly in x £ [0,l].
Along the lines similar to the proof of the first
assertion of theorem 2, it can be shov>m that, under the
hypotheses of theorera 3, there exists a finite subset
Jc ,c ,...,c }, say, of C such that for an arbitrary
6 > 0 there exists a positive number A. such that o
|f(t)| g A for all t satisfying |t-c | g 6 ,p=1,2,...,r
Choosing 6 sufficiently small, if a is an arbitrary
positive number, one can choose a positive number
A(a), say, such that for all p=1,2,...,m and all t e [0,1]
f(t) g A(a) expj f -^ } A* (t-c b Cp^l pi
whenever |t-c | < 6 ( t / c ) .
Let A = max JA^, f(c^), f(c2),..., fioj] .
By theorem 1, for all values of v(v=0,1,..., n) such
that |f(v/n^)| g A we have
(14) 2^ Pnv^")^^I^)| IV I . * * | ^ x | g 6
^ r^ A —6 n ^
g 2 A e — 0, as n^ -* oo , -1
uniformly in x £ [0,l], where 2 denotes that the
summation is restricted only to such values of v. 2
Also, if 2 denotes the sumraation restricted to the
remaining values of v, v/e have
131
2^ V (x) f(-^)l
* ,2 m
gA(a) 2 p^_^(x)J 2 exp _ - _ — _ }
/ ^ I r-i T> 1 ' I V ^ -X g 6 ^ • "c ^'n. p' •n^ I P *
m gA(a) 2^ p^ . I2 exp ^~
""*^ p=1 A* (A-^n,)) I — -, - -'n^ ' P P
•X g 6 ^ c ' c
for all n^ sufficiently large (using the property that
for p=1,2,...,m, c does not admit the approximation
A (v) through l). P
Using the monotonicity of A (v) and the definition °P
of A* (t), p=1,2,...,m, and choosing g > 0 to be °P
sufficiently small we have
m (15) A(a) ly p (x) J 2 exp -^ 1
V *• P=1 A* (A^ (n, \—-x\ g 6 ^ • "c ^"c ^"*' 'n^ I " P P
g(n^+i)
g m A(a) e ^ Pn v "*
g(n^+l)-62S g 2m A.'a) e
-• 0, as n^ -* oo ,
uniformly in x £ [0,l]. Thus (14) and (15) together
imply that
I 2 Pn v "" ^^^^1 ^ °' a^ n* ^°° '
uniformly in x £ [0,l]. This completes the proof of
theorem 3.
THEOREI'I 4 Let C be a collection of rational points c
132
lying in [0,l] and let Jp } , n=1,2,..., be a
monotonically increasing sequence of positive integers
diverging to infinity with n, such that for each c £ C
all but a finite number of cp , n=1,2,..., are integers,
Let f be a function defined on [0,l] possessing the
property that if Jt, } is any sequence of real numbers
such that lf(t, )| -* oo as k -» oo then Jt, } has a limit
point c £ C and that for each c £ C and a > 0
(16) f(t) = 0(exp - p ^ ), as t - c, (t ^ c).
Then for each x £ [0,l] where f is continuous there
holds
(17) lim B^ (x) = f(x). n -• oo - n
Also (ll) holds uniformly in x £ [a,b] S. [0,1 ], if f
is continuous at each x £ [a,b],
On the other hand, if f is a non-negative function
such that for some c £ C and for some g > 0 there holds
(18) exp - r ^ = 0(f(t))
as t -• c, (t / c), then
(19) B (x) - 00 , as n - 00 , n
for every x satisfying One of the inequalities
(20) c e~" < X < c
or
(21) c < X < 1 - e~°'(l-c).
Proof Under the given assumptions, for each c £ C, there
holds
inf I - cl g -^ ' p ' p
0 g V g P n - n 1 n
V f cp
133
provided n is sufficiently large. Thus if v/e forbid v
to assume +- e -"-alues cp (n=1,2,...) and define
A(V) = l/v(v > O) then c does not adi.iit the approximation
A(V) through Jp ). Keeping this in mind, rest of the
proof of the first assertion of tneorem 4 follov/s along
the lines of the proof of theorem 3.
To prove the second assertion, it is sufficient
to show that for one of the values of v satisfying
v/p -c| = l/p , v/p e (0,1) and for each x
in the asserted range the sequence
(22) J(^^")x^1-x)^^-^e^^"j , (n=1,2,...),
diverges to i n f i n i t y with n .
The case when c=0 or 1 presents no d i f f i c u l t y and
can be proved r a t h e r e a s i l y . So v/e consider the case
v/hen c £ ( 0 , l ) . Using S t i r l i n g ' s formula
P +i -P Pn. ~ y ^ Pn^ e ^
' ^ , v ^ , - v ^ ^ ^ ^ _ ^ ) - n - v ^ - , - ^ - n - v )
Pn^^ Pn
,r^ v+-i f .Pn-^^S-• n
p -V - 1 ^ ) ^ i - : ^ ) " )
V2iiv(1-cJ ' ~Pn' ' Pn
Hence P P -V P a
C) x ^ 1 - x ) ' - \ ' -V
Pn Pn
134
Let X satisfy one of the conditions
c > X > c e~" or c < X < 1-e~"(l-c).
Then in the first case
1 > J> e-« a n d ^ > 1,
and in the second case
X . . , 1-x V -a
- > 1 and yi^ > e .
Thus, in each case, if n is sufficiently large, we have P p -V P a /• nx v/-, \ n - n (_. ) x''(l-x) e
1-E o(p -maxlp -v,vl)
V27ic(1-c)p^
l-Ej g max Jp^-v,v}
e V'27ic(1-c)p
whore e is a positive number depending on n such that
e -•O asn-*oo. Since n
_i a max Jcp^,(1-c)p^} p " e •^n
as n -• oo , the required divergence of (22) follo\/s.
This completes the pr.. of of theorem 4.
THE0REI4 5 Let I be an unbounded set of positive
integers, A(v) a function as in definition 1 and let
c £ [0,1] be such that it admits the approximation A(v)
through I. Let f,g be tv/o non-negative functions
defined on [0,l] such that for some a > 0
(23) exp f - T - = 0(f(t)), as t - c, (t / c), A*(|t-c|-^)
and
(24) exp - r ^ = 0(g(t)), as t - c, (t / c).
135
Then for each x satisfying one of the inequalities
c e"" < X < c
or c < X < 1 - e""(l-c) ,
at least one of the sets
jB^(x), n £ 1} and JBJ(X), n £ Ij
are unbounded. Moreover, if c is irrational then for all
X satisfying one of the above inequalities
JB ( X ) , n £ Ij is alv/ays unbounded.
Proof. Since c admits the approximation A(V) through I,
there exists an infinite sequence I*, say, consisting
of elements of I such that for each n £ I* there
exists a positive integer v such that
|c - 1 g A(n).
Let I. be the subsequence of I* such that for each n £ I
there holds
0 < |c - Jl g A(n)
for some positive integer v. Also let I denote the
subsequence of I* such that for each n £ I there holds o
c = ^
n
for some integer v •
It is clear that at least one of I and I. is an o 1
infinite sequence. Moreover, if c is an irrational then
I must be infinite.
Consider the case v/hen I is infinite. V/hen c 0
coincides with one of the points 0 or 1 the proof is
relatively easy. So we consider the case when c £ (0,1).
To each n £ I , choose v such that c = v/n. To shov/ the
unboundedness of jB^(x), n £ Ij it is sufficient to prove 136
that - ^
def ( n )^v+1^^_^)n-(v+l) ^ n
as n -» oo through I . We have
1-c X wn! x^ (1 --X) ~^ .gn^ 'n ^c+1/n 1-x' ^ v!(n-v)'
^oc^ w /• T"C X ^/n! x u-x; gnx (25) W^ = (—77- j—){ ..,/ 1 \ / e ).
Frora the proof of the second assertion of theorem 4
it follows that the second factor on the right hand side
of (25) diverges to infinity v/ith n for each x lying
in the given range. Since the first factor of the same
expression converges to a positive limit, the required
divergence of V/ follows,
When I- is an infinite sequence, to each n £ I.
let V he a positive integer such that
0 < Ic - -1 s A(n). ' n' ^
I t follov/s t h a t
A*(|c- jr^) g A*(A-^n)) = ^ .
Thus
exp 7— g exp g(n+l).
A*(ic-jr') Hence to shov/ that jB (x), n £ Ij is unbounded, it is
sufficient to prove that for the given range of x
„ def /nx \),. \n-v a(n+1) V = ( ) X (1-x) e ^ ^- 00 n v
as n -• 00 . This again is evident from the proof of the
Second assertion of theorera 4.
This completes the pr -of of theorem 5.
137
CHAPTER 4
APPROXIMATION OF UNBOUNDED FUNCTIONS BY OPERATORS OF
INTEGRAL TYPE
For linear positive operators of integral type, i.e.
those defined by means of an integral such as
L(f;x) = I f(t) K(t,x)dt ,
L
where K(t,x) is a non-negative kernel and D denotes
the range of integration, integrability of the function
f is demanded by the nature of the definition of L.
For sequences of linear positive operators of integral
type (e.g. the Gamma operators) unboundedness at the
extremities of the range of integration is often seen
to be permissible for the functions to be approximated at
interior points of continuity. Hov/ever, in such cases it is
generally assumed that the functions are bounded in the
intermediate range (collection of all compact subsets
contained in the interior of the closure of the range
of integration). In the sequal we show that, with an
asymptotic analysis of the kernels of a sequence of
linear positive operat^-rs, generally it is possible to
drop down the assumption on_the boundedness of the
functions in the intermediate range. V/e limit our
attention to four operator sequences, viz. the generelized
Jackson operators and the Vallee-Poussin's integrals,
the Gamma operators of Muller and the singular integrals
of Gauss-Weierstrass v/hich provide approximations for
functions defined on a finite interval, a semi-infinite
interval and the whole real line, respectively.
138
4.1 Approximation of unbounded integrable functions
4.1.1 The general method
Let JL , n £ uj be a class of linear positive
operators of integral type and let JK (t,x), n £ Uj
be the class of the corresponding kernels where t and
X range over the subsets X and X of R, respectively.
Assuming that for each fixed n and x, K (t,x) regarded
as a function of t is essentially bounded and measurable
on X the operators L are defined for all functions n
integrable on X. The general method in the following
four cases consists in showing that
(1) Tim sup jK ( t , x ) } = 0
n -»oo t £ X ~ J s : | s -x | < 6 }
for each 6 > 0 and x under consideration. V/hen (l)
holds the basic convergence result can be shov/n to hold
for the unbounded functions provided it holds for the
bounded functions. When we pass on to asymptotic
estimates and asymptotic formulae etc. we also require
an estimate of the speed of convergence of (l).
V/e remark that v/hile dealing with the operators
of integral type it is not necessary to assume the
"strict definiti-/ns" of continuity and differentiability
etc. of the functions to be approximated. Thus, for
instance, if given an arbitrary e > 0 it is possible
to find a 6 > 0 such that
|f(t) - f(x)| < e
for "almost all" t £ (x-6, x+b), then for our purposes
the function f is "continuous" at the point x. The
results in the sequel may be considered in the light
of this slight generality.
139
4.1.2 The generalized Jackson operators L _
For n,p positive integers and x £ R, the operators
L are defined by np-p
(1) L (f;x)=^-^ r f(x+t) (^iM^)2P dt, ^ ' np-p^ ' k \ ^sm i-t ' ' ^ ^ np-p - "
—71
where
71
A = f (^il^)2P dt , np-p j s m gt'
-71
and f is a real or complex valued function integrable
on the interval [x-7t, x+7t]. For the functions f which
are 27t-periodic and integrable on [-71,71] , L (fjx)
reduces to a trigonometric polynomial of degree atmost
np-p. Por bounded functions which are 27i-periodic and
integrable on [-7[,7i]approximation properties of the operators L have been extensively studied. Some
np-p
pertinent references are ([62],[63],[39],[26] and [55])«
LEIH'IA 1 If f is a 271-periodic function integrable on
[-71,7t] , then for an arbitrary positive 6 < TI
. nt
(2) ^ ^ — f |f(x+t)| { - ^ ^ )2P dt
P-P 6 g hi g 7. "^^2
g A n^"^P sin "2P I , (n,p=1,2,...)
v/here A is a positive number depending on f but indepen
dent of n,x and 6.
Proof V/e have
nt n
r |f(x+t)| (^^^^)2P dt g sin-2P I r|f(t)|dt. 6g'^|t|g7; ^^^2 -\
140
Putting A = A ||f(t)|dt, where A is a number
-71
satisfying
71 . nt s m (3) r ( f )2P dt g A n^P-^ ,(n=1,2,...),
• sin — -7t 2
the inequality (3) follov/s. A proof of the existence of
a number A satisfying (3) is given in [62], p. 51*
THEOREM 1 Let f be a 27t-periodic function integrable
on [-71,7t] such that at a point x both f(x+) and f(x-)
e x i s t . Then f o r p = 1 , 2 , . . .
r A\ n- T /• ^ \ f (x+) + f ( x - ) (4 ) Ixm L ( f ; x ) = ^ ^ '
n — oo
Further, if f is continuous at each x £ [a,b]
S:[-7i,7t] then
(5) lim L (f;x) = f(x) ^ ' np-p^ ' ^ ' n -* CO
uniformly in x £ [a,b] .
Proof. If f is bounded the results (4)-(5) follow from
a theorem of Schurer [62], a theorem of Korovkin [30]
and the symmetry of the positive kernels (sin -p-/sin -xj'
The general case is easily proved from this particular
case and an application of lerama 1 with 6 sufficiently
small.
In a similar way we obtain the followihg
generalization of a theorem of Schurer [62].
THEOREM 2 Let p g 2. If f is a 27[-periodic function
integrable on [-71,71] and tv/ice differentiable at a point
X, then
141
(np-p)
(6) L (f;x)-f(x) = (1 - -J ^)f"(x) + o ( ^ ) , ^ ' np-p^ ) / \ / X (np-p) ^ ' V 2 "
^o ( ^ \
(n - ooj
where the coefficients y P P and p are given by
the identity . nt
/7^ /^^" ~ N 2 P 1 (np-p) 1 "P-P (np-p) (7) (—T") - 6 Po + 3 ,^ p ^ ^^ cos kt,
sxn — k=1
valid for all t.
Further, if f"(x) exists in an open interval
containing the interval [a,b] and is continuous at each
X £ [a,b], then (6) holds uniformly in x £ [a,b].
V/e remark that the theory of linear combinations of the opero-tors L given in [551 can also be extended ^ np-p ' L^-^j
by lemma 1. Thus all the results of [55] are valid for
functions v/hich are integrable but not necessarily
bounded. A similar remark holds for various other known
results on the speed of conver;-;ence of L (f;x) to • ^ np-p^ ' '
f(x).
4.1.3 The Gairima operators G
Let A be the space of all complex-valued functions
f v/hich are measurable in (0,oo ), bounded in any interval
[Y,R] , 0 < Y < R < <», and such that f(t) = 0(e^'^) as
t — 0 and f(t) = 0(t ) as t — oo for some positive
constants a and b. The n-th Gamma operator for f £ A is
defined by ([45])
-1 oo
n+1 r. (1) G (f;x) = 2 L _ ^ ^-^ f(Il)du , X > 0, \ / n^ ' n! J ^u' ' '
0
n=1,?,..., which for a fixed x exists at least for
142
n g max J[J] + 1, [b]j.
Approximation properties of the operators G are studied
in [45],[46],[47][32] and [35]; very recently the
author has obta-ined Nikolski constants, general linear
combinations and general simultaneous approximation
properties of the operators G .
Dropping the boundedness assumption, we define
A* to be the class of all complex-valued functions f
which are measurable in (0,oo ) 3,nd are such that
f(t) = 0{e^''^) as t - 0 and f(t) = O(t^) as t - oo for
some positive constants a and b.
To extend the study of the operators G to the
class A* of ftmctions we require some preliminary
estimations. Ve ha,ve
(2) G (f;x) = ^^r"" f^" e-"^ f(- )du 0
vn+1 ^nx)
n+ - -n _ 0
XI - n u x „/i\-, T u e f(—Jdu -__ n+ - -n ^u^
V?Jt n "e
I ? " vl X / \n f n -nux r.A\.,
= V 2 ^ ^"^) J ^^u)'^^' 0
by an applica,tion of the Stirling's formula for the
gamma functions.
Por x,t > 0 let p(t,x) be the ftmction defined by
(?) p(t,x) = t e - e x .
We keep x fixed and regard p(t,x) as a function of t.
By elementary calculus one easily shows that p(t,x)
is non-negs.tive and assumes its minimum value only at
the point t=x. Thus for an a-rbitrary 6 > 0, the ftmction
143
p(t,x) has a positive lower bound A ( 6 ) , say, for all
t > 0 and satisfying |t-x| g 6 . Hence for |u -x| g 6 ,
where u > 0, we have
xu e_
Thus
g ex + A ( 6 ) . u ^
(4) (^)''S (ex + A(6))-^ , (n=1,2,...). e
Let 6 be a sufficiently small positive number and let
A be a sufficiently large positive number.
Put
1 1
^6,0 = t6o»Aj ~ (^^ , ),
where 6 is an arbitrary number satisfying 0 < 6 < x.
Then if f £ A*, we have
\ { ^ (ex)° / u V ° - l(l)au| I^ 6,0
(5)
-D 1 X / ex \n
= o(Xj)
n
for an arbitrary k > 0. It is readily verified that if
0 < a < b < oo, then (5) holds uniformly in x £ [a,b].
From the known properties of the Gamma operators
we know that if f £ A, then f(t) being majorized by the
functions e and t as t -* 0 and t -» 00, respectively,
for some positive constants a and b, we have for each
fixed X > 0
144
(6) ^ ^ ^ r u V ^ - ^ f(^)du = o(-L), and A
/ \n+1 r /„x Qnx) I n -nux „/1\-, r ^ \ (7) ^ - t ^ e f(-)du = o(—)
for arbitrary fixed k > 0, provided A > 0 is
sufficiently large and 6 > 0 is sufficiently small.
Also , i f 0 < 6 < a < b < A < oothen (6) and (7) ho ld
uniformly in x £ [a,b]
Since (6) and (7) depend only on the fact that
f(t) is majorized by the functions e and t in the
above mentioned sense, if follov/s that A and 6 can be ' o o
chosen so that for a given f £ A*, (6) and (7) hold for
any given x £ (0,oo ). Further 6 and A can be chosen
so tnat for a given f £ A* and a,b satisfying
0 < a < b < 00, (6) and (7) hold uniformly in x £ [a,b].
Now put
6 = f°'~)~ ^' i ) ' 0< <-)• Combining ( 5 )» (6 ) and ( 7 ) we have
LEI-MA 1 Let f £ A* and let 6 be a positive number. Let
X £ (0, 00) be a fixed point such that 6 < x. Then
/ xn+l p ^ A toN (nx) f n -nux „/1\, / 1 ^ (8) i -^^ u e f(-)du = o ( — ) , as n - 00 ,
• i for an arbitrary k > 0. Also i f O < 6 < a < b < o o , then
(8) holds uniformly in x £ [a,b].
Thus in lemma 1 we have shown that the contribution
of f(t) (f £ A*), for t > 0 satisfying ]t-x| g 6 > 0,
to the integral in (2) is of order o(n ) for an
arbitrary k > 0. Hence the asymptotic behaviour of
145
G (f;x) depends only on the behaviour of f in a small
neighbourhood of the point x. Generalizations of various
knov/n results on G (f;x) for f £ A are thus immediate n^ ' '
for f £ A*. To mention a few, we have the following
results.
THE0REI4 1 I f f £ A*, x > 0 and f i s con t inuous a t t h e
p o i n t X, t h e n
(9) lira G^( f ;x ) = f ( x ) , n - • oo
F u r t h e r , i f O < a < b < o o and i f f £ A* i s con t inuous
a t each x £ [ a , b ] , t hen (9) h o l d s un i fo rmly i n
X £ [ a , b ] .
THEOREM 2 If f £ A*, x > 0 and f"(x) exists, then
(10) lira (n-l)[G^(f;x) - f(x)] =^x^f"ix), n -• oo
Further, if 0 < a < b < o o and f"(x) exists in an open
interval containing [a,b] and is continuous at each
X £ [a,b] , then (lO) holds uniformly in x £ [a,b].
For f £ A, (9) and (lO) were proved by Muller [45].
Vi/e mention that results on linear combinations and the
simultaneous approximation property of the Gamma
operators obtained by the author hold for f £ A*. There
are to be published elsev/here.
4.1.4 Singular integrals V/ of Gauss-V/eierstrass
These are defined by
2 o oo n /, x2
n r —T-(t-x) (1) V/^(f;x) = ^ j f(t; e dt,
for X £ (-00 , oo ), n > 0 and for all those functions f
defined on R for v/hich the integral in (l) exists.
Approximation properties of the integrals V/ are well
knovm for the functions f bounded on R. For the class
146
E^ n of functions these properties (and also those of U,R the linear conbinations of W ) are studied in [54].
Let W_ „ denote the class of all functions f each bJ,R
of which is integrable on [-YIY] f°^ each y > 0 and
satisfies the growth restriction ,,2
(2) f(t) = 0(e" ) , as |t| - oo ,
for some A > 0. It is clear that for all sufficiently
large n, W (f;x) exists for all x belonging to a given
bounded subset of R for each f £ W^ „. Q,ii
LEMI-IA 1 Let f £ W^ .„ and let 6 be a positive number. y, K
If X is a fixed point on R, then n ,'+ 2 n r •-p-(t-x)
(3) ^ J f(t) e 2 dt = o ( ^ ) ,
|t-x| g 6 "
as n — oo, for each fixed k > 0. Further if B is any
fixed bounded subset of R, then (3) holds uniformly
in X £ B, for each fixed k > 0.
Proof. For all n sufficiently large, we have
2 o 2,2 n /, N2 n t -^-(t-x) |f(t)|e 2 <i = J l^(^+*)
|t| g 6
(^ -2A)6^ r p.,2 J|f(t+x)| e-2A* dt,
e ^ dt
|t-x| g 6 |t| g 6 -2 o
2 oo — _ V a f s e
where A is a positive number satisfying (2). Now, for
each fixed x £ R (and also uniforraly for each x £ B,
a fixed bounded subset of R) the integral
/ |f(t+x)| e"^** dt
147
is bounded. Since
-i^2A)b' _ ,,) e = o(n ^ ' ) , as n — oo ,
for each fixed k > 0, the lemma follows.
It is evident from lemma 1 that to an order o(n ),
where k is an arbitrary fixed positive number, the
asymptotic behaviour of W (f;x) for f £ W depends
only on the behaviour of f(t) for t lying in an
arbitrarily small neigbourhood of the point x. In
particular from the corresponding well known results
for bounded functions, we deduce.
THEOREM 1 If f £ W^ „, X £ R and f is continuous at x, Ufti
then
(4) lim V/^(f;x) = f(x). n -• oo
Further, if f is continuous at each x £ [a,b]
(-00 < a < b < oo), then (4) holds uniformly in x £ [a,b].
THEOREM 2 If f £ W. _ , x € R and f is twice differen-
tiable at x, then
(5) lim n2[w^(f;x) - f(x)] = ^ . n -* 00
Further, if f"(x) exists in an open interval containing
[a,b](-oo < a < b < 00) and is continuous at each x £ [a,b]
then (5) holds uniformly in x £ [a,b].
In a similar way lemma 1 extends other known results
on the asymptotic estimates of the quantity V/ (f;x) and
the linear combinations ([54]) of W etc.
4.1.5 De La Vallle-Poussin integrals V
These are defined by 71
(^) V " ' " ) = 2,;(2n-JJM / f ( x + t ) c o s 2 n I d t ,
148
n=1,2,..., for all functions f and points x € R for
which the integral in (l) exists. In particular, if f
is integrable on [-71,71] and is a 27i -periodic function
then V (f;x) exists for all x and n and defines a n^ '
trigonometric polynomial of degree g n.
LEMMA 1 Let f be a 27t-periodic function integrable
on [-71,7i] . If 6 < 71 is an arbitrary fixed positive
number, then
(2) 2 i { i ^ [ If(-+*)! -^'" I ^^ = °(i^)' K g |t| g 6
as n -• 00, for each fixed k > 0, uniformly for all x £ R.
Proof. We have 7t
r if(x+t)i cos^'' I dt g cos^" I r i f ( t ) |d t . TC g | t | g 6 - n
Since
and
( 2 n ) ! ! „/ -i-N (2n- l ) ! ! = °(^ ) ' "
2n 6 „ / -(k+4-)N cos r- = o ( n '^') , n -* 00
for an arbitrary fixed k > 0, (2) follows.
Using lemma 1, from the corresponding well knov/n
results on the approximation of bounded functions by the
integrals V ([51], [62]), in particular, we deduce
THEOREM 1 Let f be a 27i-periodic function integrable
on [-71,7i] . If f is continuous at a point x £ R, then
(5) lira V^(f;x) = f(x). n -• 00
Further, if f is continuous at each point x of an inter
val [a,b], then (3) holds uniforraly in x £ [a,b].
149
THEOREM 2 Let f be a 27t-periodic function integrable
on [-71,Tt]. If f"(x) exists at a point x £ R, then
(4) lim n[V^(f;x) - f(x)] = f"(x). n -» oo
Further, if f"(x) exists at each x belonging to an
open interval containing an interval [a,b] and is
continuous at each x £ [a,b], then (4) holds uniformly
in X £ [a,b].
150
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Doctoral Thesis, Technische Hochschule Aachen,1970,
85 p.
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159
Samenvatting
Deze dissertatie behandelt de bepaling van klassen van
onbegrensde functies, die met behulp van lineaire posi-
tieve operatoren, behorende tot een gegeven verzameling,
kunnen worden geapproximeerd. Zulk een bepaling wordt
mogelijk gemaakt door het gedrag van deze operatoren
te onderzoeken t.o.v. zekere functies die tot de verza
meling van testfuncties of begrenzende functies behoren.
Een andere toegang tot dit probleem is het analyseren
van de kernen van de operatoren, indien informatie viae-
nige begrensde functies resulteert in een bepaling van
approximeerba,re onbegrensde functies.
Hoofdstuk 1 bespreekt de methode der testfuncties
en heeft het doel hun onbegrensdheid optimaal te ge-
bruiken teneinde approximeerbare onbegrensde functies
te bepalen, die een vergelijkbare orde van onbegrensdheid
bezitten. De fundamentele convergentie- en asymptotische
formules worden afgeleid; zij geven een nauwkeurige
graad van approximatie.
In hoofdstuk 2 wordt de methode der begrensde func
ties bestudeerd; het doel is meer omvattende klassen van
approximeerbare functies te bepalen, dan door anderen
verden verkregen. Hier bestaat de techniek eveneens uit
het benutten van de voile omvang van het onbegrensd
zijn van de testfuncties.
Hoofdstuk 3 behandelt de analyse van de kernen van
operatoren van het sommatietype. In het geval van de
Bernstein operatoren worden zekere klassen van onbe
grensde functies verkregen, die met behulp van deze
operatoren of deelwijze daarvan approximeerbaar zijn.
Hoofdstuk 4 analyseert op overeenkomstige wijze de
160
de kernen van operatoren van het integraaltype. Er
wordt aangetoond dat met velen daarvan niet-begrensde
functies in een punt, waar zij continu zijn, kunnen
worden geapproximeerd. Wij beperken ons daarbij tot de
generaliseerde operatoren van Jackson, de integraal-
operatoren van De La Vallee-Poussin, de operatoren,
gevormd door raiddel van de singuliere integralen van
Gauss-Weierstrass en de Gammaoperatoren van Muller.
161
Stellingen
Let JL j(n=1,2,...) be a sequence of variation
diminishing linear operators mapping P -» P and /• '\ f -r>\ m ra
C ' [a,b] - C [a,b] (m,y-0,1, 2,...), v/here P is the space of all algebraic polynoraials of degree g m and
C ' [a,b] i"' the space of all r-times continuously differentiable functions defined on [a,b].
F o r each f £ C^''^[ a , b ] , | | L ^ ^ ^ f ) - f < '' | |(j[-a,b]-^0 " n -• oo i f and on ly i f | | L ( t ; x ) - x M r -i -• 0 a s n -* oo
n (_/[_ a , DJ
for 1=0,1,2. Here r takes any value 0,1,2,... .
For an arbitray j;*-0,1 ,2, .. . for each f £ C ' [a,b] and
X £ [a,b] such that f " (x) exists v/e have
L^^)(f;x) - f^^)(x) = p4^J(r-l)(r-2)a^-2.?(r-2)a2
+ r(r-l)a4 f ' (x) + J2x[(r-l)a^-(2r-l)a2+ra^]
+ a,+ra^}f^^"^ - (x) + Jx (a^-2a2+a^)+x(a -2aj)
+ a^]f^''^^hx)] + o ( ^ ) , (n-~)
where (p(n) j^ 0, (fin) -*oo, n-*oo and a., 1 = 1,...,6 are
constants if and only if a. ^
L (l;x) = 1 + -7-^- + o(-7-^)
L^(t;x) = X + ^/,/ + o{^j^) (n - oo )
„ „ a X +acX+a^ . T f J.2 ^ 2 n 5 6 ^ 1 • L (t ;x) = X + 7^ + (—7~v n ' ' 9(n) l)(n)'
and for some p o s i t i v e i n t e g e r m
>2m+2 ^ /• 1 _ I [ t - x n L„( ( t -x) ^ ;x) = o ( ^ ^ ) , ( n - o o )
I I
A linear operator L has total variation diminishing
property for all f, i.e. T[L(f)] g T[f], if and only
if it has such ^.joperty for 'all monotone functions.
I l l
Let A,B be subgroups of a finite group. Let A v B denote
the subgroup generated by all the elements of A a-nd B.
Let the symbol [G] denote the order of a group G. Then
if and only if AB = BA.
IV
Let 0 < a < p < oo and let L be a linear operator
defined on a d imain of continuous functions defined on
an interval. Given a function g(t) define
g(a) , t < a
&-^(t) = I g(t) , a g t g p g(p) , t > p .
Por each f there holds
T[L(f)P] g T[fP] a-' •- a-
if and only if L is variation diminishing and L(l)=1.
The "if" part of this assertion is a result of I.J.
Schoenberg,
V
Let L be a non-trivial (i.e. not of the form L(f)
= ()i(f)T(x) v/here (|) is a functional and f a fixed
function) linear operator. L is said to be VD (variation m ^
diminishing of order m( if v(L(f)) g v(f) v/henever v(f) g m, (v(f) denoting the number of sign changes of the function f). L is said to be VD on a function m
' m ^
if v(L(f)-(p) g v(f-(p) v/henever v(f-(p) g m. L is VD on q)
if and only if L is VD and (p is a fixed point of L.
4
VI
For p=1,2,... let (L,p) be the summability method
defined by the matrix (a (p)) (n,v=0,1 , 2,,..) v/here
a (p)=1, a (p)=0, v=1,2,... and for n=1,2,... and 00 Ov
v=0,1,2,... a_(p) = c_(p)/^,(^'P)
where
and
nv nv 0
,,(n,p) _ ^~ ( -,\Jr2P I'np+p-nj-ls
^ " jfo j 2p-i ^ • (L,P) is a strongly regular summation method for
p=1,2,.... If f is a 27t-periodic integrable function
possessing the r derivatives f '^(x+), f (x-)
(r=0,1,2,...) at a point x, then S^ (x) the r
differentiated Fourier series of the function f is
(L,p) summable to the value (f ^Hx+)+f ^^x-) )/2 if r+2
p g """p™. Here, as is usual a derivative of order zero is synonymus with the function.
VII
Let f be a complex valued function defined on (O, oo)
and satisfying
|f(t)| < A(e«* + e«/^) , t £ (0,00 )
where A and a are two positive numbers. Let I (z) be th ^
the n modified Bessel function z" izZ2£ -2
f J x ) = f(]j (f)= + 1 + | ) , X £ (-00 ,oo )
Let
and define
Jjf;x) = exp J-n(x+^)j I i (2n)x-fJ^) v=-<»