Applied Mathematics and Computation 215 (2010) 3271–3279

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

The M-band cardinal orthogonal scaling function

Guochang Wu a,*, Dengfeng Li b, Huimin Xiao a, Zhanwei Liu c

a College of Information, Henan University of Finance and Economics, Zhengzhou 450002, Chinab College of Mathematics and Information Science, Henan University, Kaifeng 475001, Chinac School of Information Engineering, Zhengzhou University, Zhengzhou 450001, China

a r t i c l e i n f o

Keywords:Sampling theoremCOSFWaveletFilter coefficientSymmetry property

0096-3003/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.amc.2009.10.015

* Corresponding author.E-mail address: archang-0111@163.com (G. Wu)

a b s t r a c t

The M-band symmetric cardinal orthogonal scaling function with compact support is ofinterest in several applications such as sampling theory, signal processing, computergraphics, and numerical algorithms. In this paper, we provide a complete mathematicalanalysis for the M-band symmetric cardinal orthogonal scaling function. Firstly, we gener-alize some results of the cardinal orthogonal scaling function from the special case M ¼ 2to the most general case M P 2. Also, we find some new results. Secondly, we obtain thecharacterizations of the M-band symmetric cardinal orthogonal scaling function and revisitsome known examples to prove our theory.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

The sampling theorem plays a crucial role in many fields such as signal processing, image processing and digital commu-nications: it tells us how to convert an analog signal into a sequence of numbers, which can be processed digitally or becoded on a computer. For a band-limited signal, the classical Shannon sampling theorem [1] provides an exact representa-tion by its uniform samples with sampling rate higher than its Nyquist rate.

In the classical Shannon sampling theorem, the interpolant is the modulated sinc function. The sinc function also plays arole of a special scaling function from a multiresolution analysis point of view [2]. Therefore, the sampling theorem was nat-urally extended to wavelet subspaces by Walter [3]. From then on, there exist many surprising results.

Walter gave a sampling theorem describing the reconstruction of a function f ðxÞ in a scaling space from its samples. Heshowed that, if a signal f ðtÞ is in the multiresolution space Vjðj 2 ZÞ, the equation

f ðxÞ ¼X

n

fn

2j

� �vð2jx� nÞ ð1:1Þ

holds, where the interpolant vðxÞ has its Fourier transform

vðxÞ ¼ uðxÞPnuðxþ 2npÞ ð1:2Þ

and uðxÞ ¼R

R uðxÞe�ixxdx is the Fourier transform of scaling function uðxÞ andP

nuðxþ 2npÞ – 0 for any real x. This the-orem does not require that the scaling function be cardinal (see below), i.e., the interpolant is generally not the same functionas the scaling function. Later, Janssen [4] extended Walter’s result to the uniform non-integer sampling which is also calledshift sampling by Zak transform.

. All rights reserved.

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3272 G. Wu et al. / Applied Mathematics and Computation 215 (2010) 3271–3279

In paper [5], Xia and Zhang considered the case in which uðxÞ is an orthogonal scaling function satisfying the propertyuðnÞ ¼ d0;nðn 2 ZÞ. They called it a cardinal orthogonal scaling function (COSF). They classified COSF and proved that a scalingfunction uðxÞ with compact support is a COSF if and only if uðxÞ is the Haar function. Furthermore, Wu et al. [6] gave somenew characterizations about COSF.

Thus, the Haar system is the only two-band orthogonal compactly supported system that possesses cardinal property.Also, Daubechies [7] proved that up to an integer shift and possible sign change, except for the Haar wavelet, thereexists no dyadic orthonormal wavelet which is symmetric or antisymmetric about some point on R. But it is well knownthat the symmetry of the function is very important in application. The compact support of the function is a desiredproperty, too.

It is very natural to consider using a dilation factor other than 2. In [8], Chui and Lian considered construction ofsymmetric and antisymmetric orthonormal wavelet with dilation factor 3. In [9], Han constructed symmetric and anti-symmetric orthonormal wavelet with dilation factor 4. In [10], Wang constructed new symmetric scaling functions andintroduced the Batman family of continuous symmetric scaling functions with very small supports. But few people dis-cuss their cardinality. Recently, two-band interpolating multiwavelet system has been much attention [11–14] due tothe fact that the cardinal multiscaling functions can simultaneously possess such as compactly supported, orthogonal-ity, symmetry and high approximation order. Naturally, they obtained sampling theorem in the multiwaveletsubspaces.

The aim of this paper is to provide a complete mathematical analysis for the M-band symmetric cardinal orthogonal scal-ing function. Of course, in general case, the situation is different. Firstly, we generalize some results of the cardinal orthog-onal scaling function from the special case M ¼ 2 in [5,6] to the most general case M P 2. Also, we find some new results.Secondly, we discuss the characterization of the M-band symmetric cardinal orthogonal scaling function and revisit someexamples to prove our theory.

Let us now describe the organization of the material that follows. Section 2 is of a preliminary character: it contains var-ious results on the M-band multiresolution analysis and M-band wavelet. In Section 3, we give a complete characterizationfor the M-band cardinal orthogonal scaling function from filter coefficient and the corresponding frequency response. In Sec-tion 4, we discuss the symmetry property of the M-band orthogonal scaling function and classify the M-band symmetric car-dinal orthogonal scaling function.

2. Preliminaries

In this section, we introduce some notations and some results that we will be used later in this paper.Throughout this paper, we use the following notations. R and Z denote the set of real numbers and the set of integers,

respectively. L2ðRÞ is the space of all square-integrable functions, and h�; �i and k�k denote the inner product and norm inL2ðRÞ, respectively, and lðZÞ denotes the space of all square-summable sequences. In this paper, a scaling function u is alwaysassumed orthogonal, real and piecewise continuous.

In the following, we will give various results on the M-band multiresolution analysis and M-band wavelet. People canrefer to the book [15] to know further theory of M-band wavelet.

Definition 1. A sequence of closed subspaces fVjgj2Z in L2ðRÞ is a multiresolution analysis of L2ðRÞ (MRA) if it satisfies thefollowing conditions:

� Vk � Vkþ1, for all k 2 Z;� f ðxÞ 2 Vk if and only if f ðMxÞ 2 Vkþ1, for all k 2 Z; M 2 Z and M P 2;�T

k2ZVk ¼ f0g andS

k2ZVk ¼ L2ðRÞ;� there is an element u 2 V0 such that fuð� � lÞgl2Z is an orthonormal basis of V0.

By the definition of multiresolution analysis above, u satisfies a dilation equation (or sometimes, we call it refinable equa-tion) of the form

uðxÞ ¼Xk2Z

hkuðMx� kÞ: ð2:1Þ

By taking the Fourier transform on the two side of (2.1), we obtain

uðxÞ ¼ HxM

� �u

xM

� �; ð2:2Þ

where

HðxÞ ¼ 1M

Xk

hke�ikx: ð2:3Þ

G. Wu et al. / Applied Mathematics and Computation 215 (2010) 3271–3279 3273

A necessary condition for orthogonality is

XM�1

s¼0

H xþ 2spM

� ���������2 ¼ 1; ð2:4Þ

this is equivalent to

Xk2Z

hkhk�Ml ¼ Md0;l: ð2:5Þ

Furthermore, fhkgk2Z satisfies:

Xk2Z

hk ¼ M ð2:6Þ

For any orthogonal MRA with scaling function u, there exist the functions wiðxÞð1 6 i 6 M � 1Þ such that the system

fwiðx� kÞ : i 2 Z;1 6 i 6 M � 1; k 2 Zg;

forms an orthonormal basis of W0 ¼: V1 � V0.Since the functions wiðxÞð1 6 i 6 M � 1Þ 2 V1, we easily get

wiðxÞ ¼Xk2Z

gikuðMx� kÞ; ð2:7Þ

where the functions wiðxÞð1 6 i 6 M � 1Þ 2 V1 are called the multiwavelet functions.By taking the Fourier transform on the two side of (2.7), we obtain

cwiðxÞ ¼ Gi xM

� �u

xM

� �; ð2:8Þ

where

GiðxÞ ¼ 1M

Xk

gike�ikx: ð2:9Þ

3. The M-band cardinal orthogonal scaling function

It is clear that, for an M-band cardinal orthogonal scaling function uðxÞ, the standard sampling theorem

f ðxÞ ¼X

n

fnM

� �uðMx� nÞ; 8f ðxÞ 2 V0ðuÞ

holds. In order to obtain the sampling theorem in the wavelet subspaces, it is sufficient to classify the cardinal orthogonalscaling function.

Thus, in this section, we will be devoted to classifying the M-band cardinal orthogonal scaling function.Let uðxÞ be an M-band COSF. Suppose the scaling function uðxÞ and the sequence fhkgk2Z satisfy (2.1). Then, we have

uðnÞ ¼Xk2Z

hkuðMn� kÞ: ð3:1Þ

Since uðnÞ ¼ d0;nðn 2 ZÞ, we have

uðnÞ ¼ hMn:

Thus, we get

h0 ¼ 1; hMk ¼ 0; for k – 0; k 2 Z: ð3:2Þ

Conversely, if the sequence fhkgk2Z satisfies (3.2), we can get that

H�ðxÞ :¼Xk2Z

hMke�ikx ¼ 1: ð3:3Þ

From (3.1), we get

u�ðxÞ ¼ H�1M

x� �

u�1M

x� �

;

where

u�ðxÞ ¼Xn2Z

uðnÞe�inx; H�ðxÞ ¼Xk2Z

hke�ikx:

3274 G. Wu et al. / Applied Mathematics and Computation 215 (2010) 3271–3279

According to (3.3), we have

u�ðxÞ ¼ u�1M

x� �

:

By repeating above equality, we have

u�ðxÞ ¼ u�1

M2 x� �

¼ � � � ¼ u�1

Mjx

� �:

Fix x 2 R, let j! þ1, we have u�ðxÞ ¼ u�ð0Þ. Suppose that u�ð0Þ ¼ 1, we get

u�ðxÞ ¼ 1:

That is

Xn2Z

uðnÞe�inx ¼ 1

and

ðuð0Þ � 1Þe�i0x þXn–0

uðnÞe�inx ¼ 0:

Since feinxgn2Z is an orthogonal base of the space L2½�p;p�, we have uðnÞ ¼ d0;nðn 2 ZÞ, i.e., uðxÞ is cardinal.Therefore, we get

Theorem 1. Define u�ðxÞ ¼P

n2ZuðnÞe�inx. If u�ð0Þ ¼ 1, then the scaling function uðxÞ is an M-band COSF if and only if thesequence fhkgk2Z satisfies

h0 ¼ 1; hMk ¼ 0; for k – 0; k 2 Z:

In the remaining part, we suppose that u�ðxÞ satisfies u�ð0Þ ¼ 1 for the sake of convenience.

Let M ¼ 2, so we get a result in [5]:

Corollary 1. A scaling function uðxÞ is a COSF with 2-band if and only if

h0 ¼ 1; h2k ¼ 0; for k – 0; k 2 Z:

Let £ðMÞ ¼ f. . . ;�2M;�M;0;M;2M; . . .g and RðMÞ ¼ f0;1;2; . . . ;M � 1g, where £ðMÞ is a lattice generated by M, and RðMÞis the set of representatives of £ðMÞ. In this way, the scaling filter coefficient fhkg can be represented by its polyphase com-ponents or subfilter coefficients, i.e.,

HðxÞ ¼ 1M

Xk2Z

hke�ikx ¼ 1M

Xj2RðMÞ

Xn2Z

hj;ne�iðMnþjÞx ¼ 1M

Xj2RðMÞ

e�ijxHjðMxÞ;

where hj;k ¼: hMkþj are referred to as the subfilter coefficients, and HjðxÞ ¼:P

n2Zhj;ne�inx are the polyphase components.From (2.5), we have

XM�1

i¼0

XM�1

j¼0

Xk2Z

Xn2Z

hi;khj;n ¼ Mdi;jdk;n: ð3:4Þ

If uðxÞ is an M-band COSF, by (3.2), we have

h0;0 ¼ 1; h0;k ¼ 0; for k – 0; k 2 Z: ð3:5Þ

Combining (2.6) and (3.5), we easily have

XM�1

j¼1

Xk2Z

hj;k ¼ M � 1: ð3:6Þ

Combining (3.4) and (3.5), we obtain

M ¼XM�1

i¼0

Xk2Z

h2i;k ¼ 1þ

XM�1

i¼1

Xk2Z

h2i;k:

Thus, we deduce

XM�1

j¼1

Xk2Z

h2j;k ¼ M � 1: ð3:7Þ

G. Wu et al. / Applied Mathematics and Computation 215 (2010) 3271–3279 3275

Furthermore, we have

XM�1

j¼1

HjðxÞH�j ðxÞ ¼1

M2

XM�1

j¼1

Xk2Z

Xn2Z

hj;khj;ne�ikxeinx ¼ 1M2

XM�1

j¼1

Xk2Z

Xn2Z

hj;khj;ne�iðk�nÞx

from (3.4) and (3.7), we get

XM�1

j¼1

HjðxÞH�j ðxÞ ¼1

M2

XM�1

j¼1

Xk2Z

h2j;k ¼

M � 1M2 :

As a result, we have

XM�1

j¼1

HjðxÞH�j ðxÞ ¼M � 1

M2 : ð3:8Þ

Therefore, by (3.2), we get

HðxÞ ¼ 1M

Xk2Z

hke�ikx ¼ 1M

XM�1

j¼0

Xn2Z

hj;ne�iðMnþjÞx ¼ 1M

XM�1

j¼0

e�ijxHjðMxÞ ¼ 1Mþ 1

M

XM�1

j¼1

e�ijxHjðMxÞ:

Conversely, if HðxÞ ¼ 1M þ 1

M

PM�1j¼1 e�ijxHjðMxÞ, then we easily get

h0 ¼ 1; hMk ¼ 0; for k – 0; k 2 Z:

By the previous argument, we know that the function uðxÞ is cardinal.In conclusion, we have proved the following theorem:

Theorem 2. The scaling function uðxÞ is an M-band COSF if and only if

HðxÞ ¼ 1Mþ 1

M

XM�1

j¼1

e�ijxHjðMxÞ;

where HjðxÞ ¼P

n2Zhj;ne�inx;hj;k ¼ hMkþj ðk 2 Z; j ¼ 1;2;3; . . . ;M � 1Þ and HjðxÞ satisfy (3.8).

Let M ¼ 2, so we get Theorem 1 in [5]:

Corollary 2. Let the scaling function uðxÞ and the sequence fhkgk2Z satisfy (2.1) with M ¼ 2. Then a scaling function uðxÞ is COSFif and only if

HðxÞ ¼ 12þ 1

2eHð2xÞeix;

where HðxÞ ¼ 12

Pkhkeikx; eHðxÞ ¼Pk

~hkeikx with ~hk ¼ h2kþ1; jeHðxÞj 1.

By Theorems 1 and 2, we gives some characterizations of an M-band COSF from lowpass filter coefficient and the corre-sponding frequency response. Then we will classify an M-band COSF from the relation between the highpass filter coefficientand wavelet. we deduce that there is a relation between the highpass filter coefficient and wavelet’s samples in its integerpoints.

Theorem 3. A scaling function uðxÞ is an M-band COSF if and only if any wavelet functions wjðxÞð1 6 j 6 M � 1Þ satisfy

wjðkÞ ¼ gjMk; k 2 Z ð3:9Þ

and GjðxÞ :¼P

k2Zgjke�ikx – 0; 8x 2 R.

Proof (Necessity). We firstly assume the function uðxÞ is an M-band COSF. By (2.7), we have

wjðkÞ ¼Xn2Z

gjnuðMk� nÞ; 1 6 j 6 M � 1:

According to uðnÞ ¼ d0;nðn 2 ZÞ, we obtain

wjðkÞ ¼ gjMk; k 2 Z; 1 6 j 6 M � 1:

Sufficiency: Again by (2.7), we have

wjðnÞ ¼Xk2Z

gjkuðMn� kÞ:

By taking the discrete Fourier transform on the two side, we have

3276 G. Wu et al. / Applied Mathematics and Computation 215 (2010) 3271–3279

Xn2Z

wjðnÞe�inx ¼X

n

Xk2Z

gjkuðMn� kÞe�inx;

then

Xn2Z

wjðnÞe�inx ¼Xn02Z

uðn0Þe�1Min0x

Xk2Z

gjke�

1Mikx:

Let

^wj�ðxÞ ¼

Xn2Z

wjðnÞe�inx; GjðxÞ ¼Xk2Z

gjke�ikx;

u�ðxÞ ¼Xn02Z

uðn0Þe�in0x;

we get

^wj�ðxÞ ¼ Gj 1M

x� �

u�1M

x� �

:

When Eq. (3.9) holds, we have

Gj 1M

x� �

¼Xk2Z

gjke�

1Mikx ¼

Xk02Z

gjMk0

e�ik0x ¼ ^wj�ðxÞ:

Because of the condition that GjðxÞ – 0; 8x 2 R, we get u�ðxÞ ¼ 1. From the proof of Theorem 1, we haveuðnÞ ¼ d0;nðn 2 ZÞ. Therefore, we conclude that uðxÞ is an M-band COSF.

This completes the proof. h

In fact, by the above proof, we can get a weaker result:

Property 1. If there exists a wavelet function wi0 ðxÞ ð1 6 i0 6 M � 1Þ satisfying

wi0 ðkÞ ¼ gi0Mk; k 2 Z

and Gi0 ðxÞ :¼P

k2Zgi0k eikx – 0; 8x 2 R, then the scaling function uðxÞ is an M-band COSF. Furthermore, we have that other any

wavelet functions wjðxÞ ð1 6 j 6 M � 1; j – i0Þ satisfy

wjðkÞ ¼ gjMk; k 2 Z:

By above discuss, we can obtain a complete characterization about the M-band cardinal orthogonal scaling function:

Theorem 4. Let the scaling function uðxÞ and fhkgk2Z satisfy (2.1), the multiwavelet functions wjðxÞ ð1 6 j 6 M � 1Þ satisfy (2.7).Then, the following statements are equivalent:

(1) The scaling function uðxÞ is an M-band COSF;(2) The sequence fhkgk2Z satisfies

h0 ¼ 1; hMk ¼ 0; for k – 0; k 2 Z;

(3) The function HðxÞ satisfies

HðxÞ ¼ 1Mþ 1

M

XM�1

j¼1

e�ijxHjðMxÞ;

where HjðxÞ ¼P

n2Zhj;ne�inx, hj;k ¼ hMkþj ðk 2 Z; j ¼ 1;2;3; . . . ;M � 1Þ and HjðxÞ satisfy (3.8).(4) Any wavelet functions wjðxÞ ð1 6 j 6 M � 1Þ satisfy

wjðkÞ ¼ gjMk; k 2 Z;

and GjðxÞ :¼P

k2Zgjke�ikx – 0; 8x 2 R.

In the following, we give two examples of the M-band cardinal orthogonal scaling function. We refer to some facts of the

paper [16].

Example 1 (Haar function). Let the uðxÞ be characteristic function of an interval [0,1], i.e.,

uðxÞ ¼ 1; x 2 ½0;1Þ;0; x 2 ð�1;0Þ [ ½1;þ1Þ:

�

It is to see, that for any natural M

uðxÞ ¼ uðMxÞ þuðMx� 1Þ þ � � � þuðMx�M þ 1Þ ¼Xn2Z

hnuðMx� nÞ;

G. Wu et al. / Applied Mathematics and Computation 215 (2010) 3271–3279 3277

where hn ¼ 1ðn ¼ 0;1;2; . . . ;M � 1Þ and hn ¼ 0 (n are other integers).Furthermore, we get

HðxÞ ¼ 1M

1þ e�ix þ e�i2x þ � � � þ e�iðM�1Þx� ¼ 1

Me�iðM�1Þx2

sin Mx2

sin x2

:

It is obvious that the Haar scaling function satisfies the property uðnÞ ¼ d0;nðn 2 ZÞ, or equivalently, the sequence fhkgk2Z

satisfies

h0 ¼ 1; hMk ¼ 0; for k – 0; k 2 Z:

thus, we have that the scaling function uðxÞ is an M-band COSF, and the sequences fhj;k ¼: hMkþjg ðk 2 Z; j ¼ 1;2;3; . . . ;M � 1Þdefined in Theorem 3 have the form

h0;0 ¼ 1; hj;k ¼ 0; for k – 0; k 2 Z:

Therefore, we can deduce that the corresponding frequency function of each subfilter HjðxÞ is 1M e�ix. In fact, the M-band

Haar wavelet can be regarded as the synthesis of M 2-band Haar wavelet.

Example 2 (Kotelnikov–Shannon scaling function). Let the uðxÞ be Kotelnikov–Shannon scaling function: uðxÞ ¼ sinðpxÞpx , then

the scaling equation (2.1) has the following form [16]:

uðxÞ ¼Xn2Z

hnsinðpðMx� nÞÞ

pðMx� nÞ ;

where

hn ¼Mpn

sinpnM:

It is obvious that the Kotelnikov–Shannon scaling function satisfies the property uðnÞ ¼ d0;nðn 2 ZÞ, or equivalently, thesequence fhkgk2Z satisfies

h0 ¼ 1; hMk ¼ 0; for k – 0; k 2 Z:

thus, we have that the Kotelnikov–Shannon scaling function uðxÞ is an M-band COSF. The sequencesfhj;k ¼: hMkþjg ðk 2 Z; j ¼ 1;2;3; . . . ;M � 1Þ defined in Theorem 3 have the form

hj;k ¼M

pðkM þ jÞ sinpðkM þ jÞ

M¼ M

pðkM þ jÞ sin pkþ jpM

� �¼ M

pðkM þ jÞ sinðpkÞ cosjpM

� �þ cosðpkÞ sin

jM

p� �� �

¼ MpðkM þ jÞ ð�1Þk sin

jM

p� �

:

4. The M-band symmetric cardinal orthogonal scaling function

In this section, we will consider the M-band symmetric cardinal orthogonal scaling function.At first, we will consider the symmetry property of the M-band orthogonal scaling function. In the following, we will give

the definition of the symmetry:

Definition 2. A function f ðxÞ is symmetric about the point a if

f ðaþ xÞ ¼ f ða� xÞ; for all x:

The function f ðxÞ is antisymmetric about the point a if

f ðaþ xÞ ¼ �f ða� xÞ; for all x:

Note that a scaling function cannot be antisymmetric, since antisymmetric functions automatically have integral zero.However, a wavelet function can be symmetric or antisymmetric.

Then, we will classify the M-band symmetric orthogonal scaling function as following:

Theorem 5. Let the scaling function uðxÞ and fhkgk2Z satisfy (2.1). Then, the following statements are equivalent:

(1) The scaling function uðxÞ is symmetric about the point a ða 2 ZÞ;(2) uðxÞ ¼ e2iaxuðxÞ; a 2 Z;(3) The lowpass coefficient satisfies:

3278 G. Wu et al. / Applied Mathematics and Computation 215 (2010) 3271–3279

hk ¼ hð2M�2Þa�k; a 2 Z;

(4) HðxÞ ¼ eið2M�2ÞaxHðxÞ; a 2 Z;

Proof. We will firstly prove that (2)) (1).Suppose that uðxÞ ¼ e2iaxuðxÞða 2 ZÞ hold.Notice that the equality uðaþ xÞ ¼ uða� xÞ is equivalent to uðxÞ ¼ uð2a� xÞ, by the definition of inverse Fourier

transform, we have

uð2a� xÞ ¼ 12p

ZRðuð2a� �ÞÞðxÞeixxdx ¼ 1

2p

ZR

ZRðuð2a� xÞÞe�ixxdxeixxdx ¼ 1

2p

ZR

ZRðuðtÞÞe�ið2a�tÞxdt eixxdx

¼ 12p

ZR

e�i2axZ

RuðtÞeitxdt eixxdx ¼ 1

2p

ZRuðxÞe�2iax eixxdx ¼ 1

2p

ZRuðxÞeixxdx ¼ uðxÞ;

where the sixth equality is obtain by the condition uðxÞ ¼ e2iaxuðxÞ.Thus, we deduce that the scaling function uðxÞ is symmetric about the point a.Secondly, we will prove that (1)) (3).Suppose that the equality uðaþ xÞ ¼ uða� xÞ hold.Then, we obtain

uðaþ xÞ ¼Xk2Z

hkuðMxþMa� kÞ ¼Xk2Z

hkuðaþ ðMxþ ðM � 1Þa� kÞÞ ¼Xk2Z

hkuða� ðMxþ ðM � 1Þa� kÞÞ

¼Xk2Z

hkuð�Mxþ ð2�MÞaþ kÞÞ ¼Xk2Z

hkþðM�2Þauð�Mxþ kÞÞ

and

uða� xÞ ¼Xk2Z

hku �MxþMa� kð Þ ¼Xk02Z

hMa�ku �Mxþ kð Þ;

so we get

hkþðM�2Þa ¼ hMa�k;

i.e.,

hk ¼ hð2M�2Þa�k:

Thirdly, we will prove that (3)) (4).If the equality hk ¼ hð2M�2Þa�k holds, we have

HðxÞ ¼ 1M

Xk

hkeikx ¼ 1M

Xk

hð2M�2Þa�keikx ¼ 1M

Xk

hkeiðð2M�2Þa�kÞx ¼ eið2M�2Þax 1M

Xk

hkeið�kÞx ¼ eið2M�2ÞaxHðxÞ:

In the end, we will prove that (4)) (2).If the equality HðxÞ ¼ eið2M�2ÞaxHðxÞ holds, we get

uðxÞ ¼Yþ1k¼1

HðM�kxÞ ¼Yþ1k¼1

eið2M�2ÞaM�kxHðM�kxÞ ¼ uðxÞYþ1k¼1

eið2M�2ÞaM�kx ¼ uðxÞeið2M�2Þax

Pþ1k¼1

M�k

¼ e2iaxuðxÞ:

As a result, we complete the proof. h

Thus, from Theorems 4 and 5, we obtain a characterization of the M-band symmetric cardinal orthogonal scaling function:

Theorem 6. The scaling function uðxÞ is an M-band symmetric cardinal orthogonal scaling function which is symmetric about thepoint a if and only if the sequence fhkgk2Z satisfies

h0 ¼ 1; hMk ¼ 0; for k – 0; k 2 Z;

and hk ¼ hð2M�2Þa�k.

Then, we will construct some examples of the M-band symmetric cardinal orthogonal scaling function.In [10], authors introduced a way to construct the M-band orthogonal scaling function as follows:

Theorem 7. Suppose that PðtÞ ¼ G2ðtÞ, or PðtÞ ¼ 1þt2 G2ðtÞ, where GðtÞ is a real-valued function and Gð1Þ ¼ 1. Then there exists a

unique the sequence fhkgk2Z such that jHðxÞj2 ¼ Pðcos xÞ.

G. Wu et al. / Applied Mathematics and Computation 215 (2010) 3271–3279 3279

Example 3. Let

GðxÞ ¼ 1þ ð1� cos xÞðc1 cos xþ c2 cos 2xÞ:

By choosing appropriate c1; c2, they deduced that the sequence fhkgk2Z has the form

. . . ;0; 0;a;12;1� a;1; . . . ;1|fflfflfflffl{zfflfflfflffl}

M�3

;1� a;12;a;0;0; . . . ;

8<:

9=;

where a ¼ 12þ

ffiffi6p

4 . They named the corresponding scaling function as the Batman scaling function.We easily see, when M is odd, the sequence fhkgk2Z is symmetric. And it is obvious that fhkgk2Z satisfy (3.2). Therefore, the

corresponding scaling function is an M-band symmetric cardinal orthogonal scaling function.

Example 4. Let M ¼ 5 and choose

GðxÞ ¼ ð1þ 4ð1� cos xÞ � 83ð1� cos xÞ2Þ2:

By Theorem 7, the sequence fhkgk2Z which has the form

115

. . . ; 0;0;�2;�2;1;6;9;16;19;16; ;9;6;1;�2;�2; 0;0; . . .f g

is the lowpass filter coefficient of a 5-band symmetric orthogonal scaling function.Notice that the sequence fhkgk2Z does not satisfy (3.2), so this 5-band symmetric orthogonal scaling function is not

cardinal.

Acknowledgements

The authors would like to express their gratitude to the referee for his (or her) valuable comments and suggestions thatlead to a significant improvement of our manuscript. This work was supported by Innovation Scientists and TechniciansTroop Construction Projects of Henan Province (No. 084100510012) and the National Natural Science Foundation of China(No. 60774041).

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