Construction of a class of compactly supported orthogonal vector-valued wavelets

Chaos, Solitons and Fractals 34 (2007) 253–261

www.elsevier.com/locate/chaos

Construction of a class of compactly supportedorthogonal vector-valued wavelets

Lei Sun *, Zhengxing Cheng

School of Sciences, Xi’an Jiaotong University, Xi’an, 710049, PR China

Accepted 30 June 2006

Abstract

In this paper, first we introduce vector-valued multiresolution analysis with dilation factor a P 2 and orthogonalvector-valued wavelet. A necessary and sufficient condition on the existence of orthogonal vector-valued wavelet isderived. Then, for a given L-length compactly supported orthogonal vector-valued wavelet system, by virtue of ans · s orthogonal real matrix M and an s · s symmetry idempotent real matrix H where M(Is � H + He�ig) is a unitarymatrix for each g 2 R, we construct (L + 1)-length compactly supported orthogonal vector-valued wavelet system. Ourmethod is of flexibility and easy to carry out. Finally, as an application we give an example.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

As we know, nature is clearly not continuous, not periodic, but self-similar. Mohamed EI Naschie with e1 has intro-duced a mathematical formulation to describe phenomena that is resolution dependent. E-infinity appears to be clearlya new framework for understanding and describing nature. As reported in [1–3], e1 space–time is an infinite dimen-sional fractal, that happens to have D = 4 as the expectation value for topological dimension. The topological value3 + 1 means that in our low energy resolution, the world appears to us if it were four-dimensional [4–6], it is shownthat the dimension changes if we consider different energies, corresponding to different lengths-scale in universe.Fourier’s transform is a mathematical tool to consider the motion in the domain of frequencies and we need a transformthat takes account into not only the resolutions but also the frequencies. The best candidate for this purpose is thewavelet transform.

Wavelet theory has been studied extensively in both theory and applications during the last decade. The main advan-tage of wavelets is their time-frequency localization property. Many signals in areas like music, speech, image and videoimages can be efficiently represented by wavelets that are translations and dilations of a single function called motherwavelet with bandpass property. Many research papers on this topic have appeared recently and most of them havefocused on scalar-value wavelets or a single mother wavelet function. It is well known that there is a limitation forthe time-frequency localization of a single mother wavelet, that is, if it is very localized in the time domain then it willnot be very localized in the frequency domain [7]. The best time-localized mother wavelet is the Haar wavelet. However,

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.06.085

* Corresponding author.E-mail address: [email protected] (L. Sun).

mailto:[email protected]

254 L. Sun, Z. Cheng / Chaos, Solitons and Fractals 34 (2007) 253–261

it is not continuous. The best frequency-localized mother wavelet function is the sinc wavelet, whose Fourier spectrumis not continuous. At the same time, it is also known that an orthogonal wavelet function with compact support andcertain regularity cannot have any symmetry [8,9]. Geronimo et al. [10] constructed two functions w1(x) and w2(x)whose translations and dilations form an orthonormal basis for L2(R). The importance for these two functions is thatthey are continuous, well time-localized (or short support), and of certain symmetry. This example tells us that if severalmother wavelets (or multiwavelets) are used in an expansion, then better properties can be achieved over single wave-lets. Further, Xia and Suter [11] show that vector-valued wavelets are a class of generalized multiwavelets and multi-wavelets can be generated from the component functions in vector-valued wavelets. Vector-valued wavelets andmultiwavelets are different in the following sense. Vector-valued wavelets can be used to decorrelate a vector-valuedsignal not only in the time domain but also between the components of vectors for a fixed time. The construction ofmultiwavelets focuses only on the decorrelation of signals in time domain. Another difference is between their discreteimplementations. Prefiltering is usually required for discrete multiwavelet transforms but not necessary for discrete vec-tor-valued transforms.

In real life, video images are vector-valued signals. Vector transforms have been recently studied for image cod-ing by Li [12,13] where input signals are finite vectors with same dimension and it has been shown that vectortransforms have advantage in image coding at low bit rates. Another type of vector transforms has been discussedin [14].

Studying vector-valued wavelets is useful in multiwavelet theory and representations of signal. In this paper,inspired by [11,15,16], we shall investigate the construction of a class of compactly supported orthogonal vector-valued wavelets. The paper is organized as follows. In Section 2, we briefly introduce the concept of the vector-valued multiresolution analysis with dilation factor a P 2. In Section 3, we discuss the existence of orthogonalvector-valued wavelets, and by virtue of the fact that if M is an s · s orthogonal real matrix and H is an s · s

symmetry idempotent real matrix then M(Is � H + He�ig) is a unitary matrix for each g 2 R, we present an algo-rithm for constructing (L + 1)-length compactly supported orthogonal vector-valued wavelet system from L-lengthcompactly supported orthogonal vector-valued wavelet system. Finally, an example is provided to illustrate ouralgorithm.

2. Vector-valued multiresolution analysis

We begin with some basic theory and notations to be used throughout this paper. Let R and C be the set of all realand complex numbers, respectively. Denote by Z and Z+ all integers and nonnegative integers, respectively. Let s be aconstant and s P 2,s 2 Z. Is and O stand for the s · s identity matrix and the zero matrix, respectively. By L2(R,Cs) wedenote the set of all vector-valued functions h(t), i.e.,

L2ðR;CsÞ :¼ hðtÞ ¼ ðh1ðtÞ; h2ðtÞ; . . . ; hsðtÞÞT : hvðtÞ 2 L2ðRÞ; 1 6 v 6 sn o

:

Here and afterwards, T denotes the transpose of vector or matrix.For h(t) 2 L2(R,Cs), its integration is defined by

RR hðtÞdt ¼

RR h1ðtÞdt;

RR h2ðtÞdt; . . . ;

RR hsðtÞdt

� �T. The Fourier

transform of h(t) is defined by hðgÞ :¼R

R hðtÞ � e�itg dt, where i2 = �1.For two vector-valued functions h(t), g(t) 2 L2(R,Cs), their symbol inner product is defined by

hhð�Þ; gð�Þi :¼Z

RhðtÞg�ðtÞdt; ð1Þ

where * means the complex conjugate and the transpose.The multiresolution analysis approach is one of the main approaches in the construction of wavelets. Let us intro-

duce vector-valued multiresolution analysis and give the definition for orthogonal vector-valued wavelets. First of all,we introduce some definitions.

A sequence {hk(t)}k2Z � X � L2(R,Cs) is called an orthonormal set of X, if it satisfies

hhkð�Þ; hlð�Þi :¼ dklIs; k; l 2 Z; ð2Þ

where dkl is the Kronecker symbol such that dkl = 1 when k = l and dkl = 0 when k 5 l.

Definition 1. We say that h(t) 2 X � L2(R,Cs) is an orthogonal vector-valued function in X if its translations{h(t � k)}k2Z is an orthonormal set in X, i.e.,

hhð� � kÞ; hð� � lÞi :¼ dklIs; k; l 2 Z: ð3Þ

L. Sun, Z. Cheng / Chaos, Solitons and Fractals 34 (2007) 253–261 255

Definition 2. A sequence {hk(t)}k2Z � X � L2(R,Cs) is an orthonormal basis of X if it satisfies (2) and for g(t) 2 X, thereexists a unique sequence of s · s constant matrices {Ak}k2Z such that

gðtÞ ¼Xk2Z

AkhkðtÞ: ð4Þ

The expansion (4) is also called the Fourier series expansion of g(t).Let U(t) = (/1(t)/2(t), . . .,/s(t))

T 2 L2(R,Cs) satisfy the following refinement equation:

UðtÞ ¼Xk2Z

P kUðat � kÞ; ð5Þ

where a 2 Z+,a P 2 and {Pk}k2Z is an s · s constant matrix sequence. If the sequence {Pk}k2Z is finite, we say that U(t)is a compactly supported vector-valued function.

Define a close subspace Vj � L2(R,Cs) by

Vj ¼ closL2ðR;CsÞðspanfUðajt � kÞgÞ; k 2 Z: ð6Þ

Definition 3. We say that U(t) in (5) generates a vector-valued multiresolution analysis {Vj}j2Z of L2(R,Cs) with dilationfactor a, if the sequence {Vj}j2Z defined in (6) satisfies the following

(1) � � � � V�1 � V0 � V1 � � � �;(2) \j2ZVj = {0}; [j2ZVj is dense in L2(R,Cs), where 0 is the zero vector of L2(R,Cs);(3) h(t) 2 V0 if and only if h(ajt) 2 Vj, j 2 Z;(4) there exists U(t) 2 V0 such that the sequence {U(t � k),k 2 Z} is a Riesz basis of V0.

The vector-valued function U(t) is called a scaling function of the vector-valued multiresolution analysis. Note that,the vector-valued multiresolution analysis which is stated in [11] or [16] is the special case of dilation factor a = 2. Weconsider the general case, that’s, dilation factor a is greater than or equal to 2 in this paper.

By taking the Fourier transform for the both sides of (5) and assuming bUðgÞ is continuous at zero, we have

bUðgÞ ¼ Pðg=aÞ bUðg=aÞ; g 2 R; ð7Þ

where

PðgÞ ¼ 1

a

Xk2Z

P k � e�ikg ð8Þ

is 2p-periodic.Let Wj,j 2 Z denote the orthocomplement subspace of Vj in Vj+1. If there exist a � 1 vector-valued functions

Wi(t) 2 L2(R,Cs),i = 1,2, . . .,a � 1 such that translations and dilations of W1(t),W2(t), . . .,Wa�1(t) form a Riesz basisof Wj, i.e.,

Wj ¼ closL2ðR;CsÞ span Wiðajt � kÞf gð Þ; i ¼ 1; 2; . . . ; a� 1; k 2 Z; ð9Þ

then we refer to {U(t),W1(t), . . .,Wa�1(t)} as a vector-valued wavelet system.For each i, from Wi(t) 2W0 � V1, there exists an s · s constant matrix sequence {Qi,k}k2Z such that

WiðtÞ ¼Xk2Z

Qi;kUðat � kÞ: ð10Þ

Let

QiðgÞ ¼1

a

Xk2Z

Qi;k � e�ikg; ð11Þ


then

bWiðgÞ ¼ Qiðg=aÞ bUðg=aÞ; g 2 R: ð12Þ

We say a � 1 vector-valued functions W1(t),W2(t), . . .,Wa�1(t) are orthogonal vector-valued wavelet functions associatedwith the orthogonal vector-valued scaling function U(t), if they satisfy

hUð�Þ;Wið�Þi ¼ O; i ¼ 1; 2; . . . ; a� 1; ð13ÞhWið�Þ;Wlð� � kÞi ¼ dild0kIs; 1 6 i; l 6 a� 1; k 2 Z; ð14Þ

and {Wi(t � k),i = 1,2, . . .,a � 1,k 2 Z} forms an orthonormal basis of W0.Thus we have

hWið�Þ;Wið� � kÞi ¼ d0kI s; i ¼ 1; 2; . . . ; a� 1; k 2 Z: ð15Þ

The following lemma, which will be used in the next section, gives a characterization in the frequency domain of anorthogonal vector-valued function h(t).

Lemma 1. Let h(t) 2 L2(R,Cs), then h(t) is an orthogonal vector-valued function if and only if
Xk2Z
hðgþ 2kpÞh�ðgþ 2kpÞ ¼ Is; g 2 R: ð16Þ

3. Construction of compactly supported orthogonal vector-valued wavelets

In this section, we first consider the existence of orthogonal vector-valued wavelets and describe the characterizationof orthogonal scaling function and orthogonal vector-valued wavelets. Then we construct (L + 1)-length compactlysupported orthogonal vector-valued wavelet system from L-length compactly supported orthogonal vector-valuedwavelet system.

Theorem 1. Let U(t) be an orthogonal vector-valued function and be defined by (5), then, for each g 2 R,
Xa�1
k¼0

Pgþ 2kp

a

� �P�

gþ 2kpa

� �¼ Is; ð17Þ

or equivalently,
Xl2Z
P lP �lþak ¼ ad0kI s: ð18Þ

Moreover, W1(t),W2(t), . . .,Wa�1(t) are the orthogonal vector-valued wavelet functions associated with U(t) if and only if,

for 1 6 i,l 6 a � 1,

Xa�1

k¼0

Pgþ 2kp

a

� �Q�i

gþ 2kpa

� �¼ O; ð19Þ

Xa�1

k¼0

Qigþ 2kp

a

� �Q�l

gþ 2kpa

� �¼ dilI s; ð20Þ

or equivalently,
Xl2Z
P lQ�i;lþak ¼ O; ð21ÞXl2Z

QiQ�l;lþak ¼ adild0kIs: ð22Þ

Proof. Suppose that, for each i, Wi(t) is the vector-valued wavelet function associated with the orthogonal vector-valuedscaling function U(t). We first show the formula (19). By (13),


O¼hUðtÞ;WiðtÞi ¼Xk2Z

bUðgþ 2kpÞ bW�i ðgþ 2kpÞ

¼Xk2Z

Pgþ 2kp

a

� �bU gþ 2kpa

� �bU� gþ 2kpa

� �Q�i

gþ 2kpa

� �¼Xk¼na

Pgaþ 2np

� � bU gaþ 2np

� � bU� gaþ 2np

� �Q�i

gaþ 2np

� �þX

k¼naþ1

Pgaþ 2p

aþ 2np

� �bU gaþ 2p

aþ 2np

� �bU� gaþ 2p

aþ 2np

� �Q�i

gaþ 2p

aþ 2np

� �þ � � � þ

Xk¼naþða�1Þ

Pgaþ 2ðn� 1Þp

aþ 2np

� �bU gaþ 2ðn� 1Þp

aþ 2np

� �bU� gaþ 2ðn� 1Þp

aþ 2np

� ��Q�i

gaþ 2ðn� 1Þp

aþ 2np

� �¼P

ga

� � Xk¼na

bU gaþ 2np

� � bU� gaþ 2np

� � !Q�i

ga

� �þP

gaþ 2p

a

� � Xk¼naþ1

bU gaþ 2p

aþ 2np

� �bU� gaþ 2p

aþ 2np

� � !Q�i

gaþ 2p

a

� �

þ � � � þPgaþ 2ðn� 1Þp

a

� � Xk¼naþða�1Þ

bU gaþ 2ðn� 1Þp

aþ 2np

� �bU� gaþ 2ðn� 1Þp

aþ 2np

� � !Q�i

gaþ 2ðn� 1Þp

a

� �

¼Xa�1

k¼0

Pgþ 2kp

a

� �Q�i

gþ 2kpa

� �:

This leads to the formula (19). Similarly, by Lemma 1 and the formula (14), we can show the formulas (17) and (20).The remaining result follows immediately.

Conversely, similar to [16, Theorem 1], we can prove that, if W1(t),W2(t), . . .,Wa�1(t) satisfy the formulas (19) and(20), then they are orthogonal vector-valued wavelet functions corresponding to the orthogonal vector-valued scalingfunction U(t). h

Now let us focus our attention on the compactly supported orthogonal vector-valued wavelet system. Suppose thatthe compactly supported orthogonal vector-valued wavelet system {U(t),W1(t), . . .,Wa�1(t)} satisfies the followingequations

UðtÞ ¼XL�1

k¼0

P kUðat � kÞ; ð23Þ

WiðtÞ ¼XL�1

k¼0

Qi;kUðat � kÞ: ð24Þ

Then the formulas (18), (21), and (22) can be written as

XL�1�ak

l¼0

P lP �lþak ¼ ad0kI s; ð25Þ

XL�1�ak

l¼0

P lQ�i;lþak ¼ O; ð26Þ

XL�1�ak

l¼0

QiQ�l;lþak ¼ adild0kI s; ð27Þ

respectively.In what follows we show that (L + 1)-length compactly supported orthogonal vector-valued wavelet system can be

constructed from L-length compactly supported orthogonal vector-valued wavelet system. Before describing the result,we give the following lemma.


Lemma 2. Suppose M is an s · s orthogonal real matrix and H is an s · s symmetry idempotent real matrix. Let

MðgÞ ¼ MðIs � H þ H e�igÞ: ð28Þ

Then, for each g 2 R, M(g) is a unitary matrix.

Proof. Since M is an s · s orthogonal matrix and H is an s · s symmetry idempotent matrix, it follows that MMT = Is

and H = HT, H = H2. By (28), we have

MðgÞM�ðgÞ ¼ MðIs � H T þ H T eigÞðIs � H T þ H T e�igÞ�MT

¼ MfIs � H T � H þ 2HHT þ ðH T � HH TÞeig þ ðH � HH TÞe�iggMT ¼ Is:

Thus M(g) is a unitary matrix. h

Lemma 2 implies that we can get a unitary matrix from any orthogonal real matrix M and symmetry idempotent realmatrix H. Next we obtain the main result of the section.

Theorem 2. Let UðtÞ ¼PL�1

k¼0P oldk Uðat � kÞ be a compactly supported orthogonal vector-valued scaling function, where

fP oldk g

L�1k¼0 is an L-length matrix sequence satisfying (25). Construct an (L + 1)-length matrix sequence fP new

k gLk¼0 as follows:

P newk ¼

P old0 MðIs � HÞ; k ¼ 0;

P oldk�1MH þ P old

k MðIs � HÞ; 1 6 k 6 L� 1;

P oldL�1MH ; k ¼ L;

8><>: ð29Þ

where M is an s · s orthogonal real matrix and H is an s · s symmetry idempotent real matrix. Then the new sequence

fP newk g

Lk¼0 also satisfies (25).

Proof. Let

MðgÞ ¼ MðIs � H þ H e�igÞ:

From Lemma 2, we know that, for each g 2 R, M(g) is a unitary matrix. By (29), we have

PnewðgÞ ¼ 1

aðP old

0 MðIs � HÞÞe�i0g þXL�1

k¼1

ðP oldk�1MH þ P old

k MðIs � HÞÞe�ikg þ P oldL�1MH e�iLg

( )

¼ 1

aP old

0 e�i0gMðIs � HÞ þXL�2

k¼0

P oldk e�ikgMH e�ig þ

XL�1

k¼1

P oldk e�ikgMðIs � HÞ þ P old

L�1 e�iðL�1ÞgMH e�ig

( )

¼ 1

aðP old

0 e�igMðIs � HÞ þXL�1

k¼1

P oldk e�ikgMðIs � HÞÞ þ

XL�2

k¼0

P oldk e�ikgMH e�ig þ P old

L�1 e�iðL�1ÞgMH e�ig

!( )¼ PoldðgÞMðIs � HÞ þPoldðgÞMH e�ig ¼ PoldðgÞMðIs � H þ H e�igÞ ¼ PoldðgÞMðgÞ:

Therefore, by (17), it follows that
Xa�1
k¼0

Pnew gþ 2kpa

� �Pnew� gþ 2kp

a

� �¼Xa�1

k¼0

Pold gþ 2kpa

� �M

gþ 2kpa

� �M� gþ 2kp

a

� �Pold� gþ 2kp

a

� �

¼Xa�1

k¼0

Pold gþ 2kpa

� �Pold� gþ 2kp

a

� �¼ Is;

which shows that fP newk g

Lk¼0 satisfies the formula (25). The proof is completed. h

Similarly, for each i, i = 1,2, . . .,a � 1, we can obtain fQnewi;k g

Lk¼0 from fQold

i;k gL�1k¼0 . So we have

Theorem 3. Let fUoldðtÞ;Wold1 ðtÞ; . . . ;Wold

a�1ðtÞg be an L-length compactly supported orthogonal vector-valued wavelet

system, and the L-length matrix sequences fP oldk g

L�1k¼0 ; fQold

i;k gL�1k¼0 satisfy (25)–(27). Construct fP new

k gLk¼0; fQnew

i;k gLk¼0 as in

Theorem 2. Then (L + 1)-length matrix sequences fP newk g

Lk¼0; fQnew

i;k gLk¼0 also satisfy (25)–(27). Consequently,

fUnewðtÞ;Wnew1 ðtÞ; . . . ;Wnew

a�1ðtÞg is an (L + 1)-length compactly supported orthogonal vector-valued wavelet system.


Proof. By Theorem 2, it follows that the sequence fQnewi;k g

Lk¼0 satisfies the formula (27), and

PnewðgÞ ¼ PoldðgÞMðgÞ; Qnewi ðgÞ ¼ Qold

i ðgÞMðgÞ:
Thus, Xa�1
k¼0

Pnew gþ 2kpa

� �Qnew�

i

gþ 2kpa

� �¼Xa�1

k¼0

Pold gþ 2kpa

� �M

gþ 2kpa

� �M� gþ 2kp

a

� �Qold�

i

gþ 2kpa

� �

¼Xa�1

k¼0

Pold gþ 2kpa

� �Qold�

i

gþ 2kpa

� �¼ O;

which implies that the sequences fP newk g

Lk¼0 and fQnew

i;k gLk¼0 also satisfy the formula (26). h

Remark 1. In Theorem 2, since the orthogonal real matrix M and the symmetry idempotent matrix H are not unique,we can construct many new compactly supported orthogonal vector-valued wavelet system.

Remark 2. For any l 2 Z+, processing the step in Theorem 2 continuously, we can construct (L + l)-length compactlysupported orthogonal vector-valued wavelet system from L-length compactly supported orthogonal vector-valuedwavelet system.

Remark 3. Naturally, one may ask how to construct L-length compactly supported orthogonal vector-valued waveletsystem from (L + 1)-length orthogonal vector-valued wavelet system? The problem is very complicated, and so far wedo not know how to do it.

From the mathematical viewpoint, the convention usually changes the mathematical properties such as regularityand approximation order. However, in what follows we will see that, in the signal processing, when the two input vectorsequences {vk},{uk} satisfy the certain linear relation, then the same output vector sequence can be produced by both

the old sequence fP oldk g

L�1k¼0 and the new sequence fP new

k gLk¼0 in the following theorem.

Theorem 4. Let two vector sequences {vk} and {uk} satisfy

vk ¼ MHukþ1 þMðIs � HÞuk ; ð30Þ

where M, H are defined in Theorem 2. Then, for any integer n,
XL
k¼0

P newk ukþ2n ¼

XL�1

k¼0

P oldk vkþ2n: ð31Þ

Proof. By (29) and (30), we can deduce that
XL
k¼0

P newk ukþ2n ¼ P old

0 MðIs � HÞu2n þXL�1

k¼1

P oldk�1MH þ P old

k MðIs � HÞ� �

ukþ2n þ P oldL�1MHuLþ2n

¼XL�1

k¼0

P oldk MðIs � HÞukþ2n þ

XL�1

k¼0

P oldk MHukþ1þ2n ¼

XL�1

k¼0

P oldk ðMðIs � HÞukþ2n þMHukþ1þ2nÞ

¼XL�1

k¼0

P oldk vkþ2n:

This leads to (31). h

4. Construction example

In this section, we give an example to illustrate our algorithm. Consider the 3-coefficient orthogonal vector-valuedwavelet system {U(t),W(t)} with dilation factor 2 constructed in [16]. Let U(t) satisfy the following equation:

UðtÞ ¼X2

k¼0

P kUð2t � kÞ;


where

P 0 ¼0 2þ

ffiffi5p

4

0 2�ffiffi5p

4

" #; P 1 ¼

2þffiffi3p

42�ffiffi3p

4

2�ffiffi3p

42þffiffi3p

4

" #; P 2 ¼

2�ffiffi5p

40

2þffiffi5p

40

" #:

and the corresponding vector-valued wavelet W(t) satisfies the following equation

WðtÞ ¼X2

k¼0

QkUð2t � kÞ;

where

Q0 ¼0 2þ

ffiffi3p

4

0 2�ffiffi3p

4

" #; Q1 ¼

� 2þffiffi5p

4� 2�

ffiffi5p

4

� 2�ffiffi5p

4� 2þ

ffiffi5p

4

" #; Q2 ¼

2�ffiffi3p

40

2þffiffi3p

40

" #:

Now we take M ¼12�ffiffi3p

2ffiffi3p

212

" #; H ¼ � 1

4�ffiffi3p

4

�ffiffi3p

4� 3

4

" #. By a simple calculation, 4-coefficient orthogonal vector-valued

scaling function Unew(t) can be written as

UnewðtÞ ¼X3

k¼0

P newk Unewð2t � kÞ;

where

P new0 ¼

2ffiffi3pþffiffi1p

58

2þffiffi5p

8

2ffiffi3p�ffiffi1p

58

2�ffiffi5p

8

" #; P new

1 ¼2ffiffi3p�3

82�ffiffi3p

8

2ffiffi3pþ3

82þffiffi3p

8

" #;

P new2 ¼

2þffiffi3p

8�2ffiffi3p�3

8

2�ffiffi3p

8�2ffiffi3pþ3

8

" #; P new

3 ¼2�ffiffi5p

8�2ffiffi3pþffiffi1p

58

2þffiffi5p

8�2ffiffi3p�ffiffi1p

58

" #;

and vector-valued wavelet

WnewðtÞ ¼X3

k¼0

Qnewk Unewð2t � kÞ;

where

Qnew0 ¼

3þ2ffiffi3p

82þffiffi3p

8

�3þ2ffiffi3p

82�ffiffi3p

8

" #; Qnew

1 ¼�2ffiffi3pþffiffi1p

58

�2þffiffi5p

8

�2ffiffi3p�ffiffi1p

58

�2�ffiffi5p

8

" #;

Qnew2 ¼

�2�ffiffi5p

82ffiffi3pþffiffi1p

58

�2þffiffi5p

82ffiffi3p�ffiffi1p

58

" #; Qnew

3 ¼2�ffiffi3p

83�2

ffiffi3p

8

2þffiffi3p

8�3�2

ffiffi3p

8

" #:

If we take M ¼ 1 00 �1

; H ¼

12

12

12

12

, then obtain the 4-coefficient orthogonal vector-valued scaling function

U0newðtÞ ¼X3

k¼0

P newk U0newð2t � kÞ;

where

P 0new0 ¼

2þffiffi5p

8�2�

ffiffi5p

8

2�ffiffi5p

8�2þ

ffiffi5p

8

" #; P 0new

1 ¼2�ffiffi5p

8�6�

ffiffi5p

8

2þffiffi5p

8�6þ

ffiffi5p

8

" #;

P 0new2 ¼

2þ2ffiffi3p�ffiffi5p

8�2þ2

ffiffi3pþffiffi5p

8

�2ffiffi3pþ25

ffiffi5p

8�2ffiffi3p�25

ffiffi5p

8

" #; P 0new

3 ¼2�ffiffi5p

82�ffiffi5p

8

2þffiffi5p

82þffiffi5p

8

" #;

and vector-valued wavelet

W0newðtÞ ¼X3

k¼0

Q0newk U0newð2t � kÞ;


where

Q0new0 ¼

2þffiffi3p

8�2�

ffiffi3p

8

2�ffiffi3p

8�2þ

ffiffi3p

8

" #; Q0new

1 ¼�6�

ffiffi3p

82�ffiffi3p

8

�6þffiffi3p

82þffiffi3p

8

" #;

Q0new2 ¼

2�ffiffi3p�2ffiffi5p

8�2þ

ffiffi3p�2ffiffi5p

8

2þffiffi3pþ2ffiffi5p

8�2�

ffiffi3pþ2ffiffi5p

8

" #; Q0new

3 ¼2�ffiffi3p

82�ffiffi3p

8

2þffiffi3p

82þffiffi3p

8

" #:

5. Conclusion

We obtain a result on the existence of orthogonal vector-valued wavelet, and construct (L + 1)-length compactlysupported orthogonal vector-valued wavelet system from L-length compactly supported orthogonal vector-valuedwavelet system. The method is easily carried out and the constructed (L + 1)-length compactly supported orthogonalvector-valued wavelets are not unique, and so we can obtain the desired one according to our need.

Acknowledgement

The authors would like to express their gratitude to the referee for his (or her) valuable comments and suggestions.

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Construction of a class of compactly supported orthogonal vector-valued wavelets

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