The research of a class of biorthogonal compactly supported vector-valued wavelets

The research of a class of biorthogonal compactly supported vector-valued wavelets

Chaos, Solitons and Fractals 41 (2009) 951–961 Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier...

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Chaos, Solitons and Fractals 41 (2009) 951–961

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos

The research of a class of biorthogonal compactly supported vector-valued wavelets Qingjiang Chen *, Ailian Huo School of Science, Xi’an University of Architecture and Technology, P.O. Box 14, Xi’an 710055, PR China

a r t i c l e

i n f o

Article history: Accepted 23 April 2008 Communicated by Prof. Ji-Huan HE

a b s t r a c t In this paper, we introduce the biorthogonal vector-valued wavelets. We prove that, like in the scalar wavelet case, the existence of a pair of biorthogonal compactly supported vectorvalued scaling functions guarantees the existence of a pair of biorthogonal compactly supported vector-valued wavelet functions. An algorithm for constructing a pair of biorthogonal compactly supported vector-valued wavelet functions is presented by means of vector-valued multiresolution analysis and matrix theory. The notion of biorthogonal vector-valued wavelet packets is introduced, and their properties are investigated by virtue of time–frequency analysis and algebra theory. Three biorthogonality formulas concerning the wavelet packets are established. Relation to some physical theories such as E-infinity Cantorian space–time theory is also discussed. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The last two decades or so have witnessed the introduction of various new fundamental theories, such as E-infinity Cantorian space–time theory [1,2], super strings [3], twistors theory [4], non-commutative geometry [5], wavelet theory, super membrane, scale relativity and loop quantum mechanics, all trying to arrive at a fully consistent, truly standard theory in which all the fundamental interactions are unified and without the need of introducing special assumptions, or putting the mass of elementary particles based on experimental observations [6–9]. At present one of the most urgent needs is to find out the number and the mass of the so-called Higgs bosons [9]. As reported by El Naschie, the standard model of elementary particles could not stand as a consistent theory without the addition of two more hypothetical particles to the 60 experimentally confirmed degrees of freedom [6]. These two particles are thought to be a massive spin boson termed the Higgs and another massless spin two particle to mediate gravity at the quantum level, named graviton. Owing to this and many other shortcomings of the otherwise singularly successful standard model there has been a flurry of attempts to extend it within the fundamental framework of quantum field theory [6,7]. Above all the new theory should make new predictions and confirm older results, experimental as well as theoretical. Mohamed El Naschie with e1 has introduced a mathematical formulation to describe phenomena that are resolution dependent. Reading carefully El Naschie’s papers [3,10–15], E-infinity space–time theory appears to be clearly a new framework for understanding and describing nature than just a new equation. When we want to consider the motion in the domain of frequencies, the mathematical tool is Fourier’s transform. The context of E-infinity is quite different, since we need a transform that takes into account not only the frequencies but also the resolutions. The best candidate for this purpose is the wavelet transform. The wavelet transform is a simple mathematical tool that cuts up data or functions into different frequency components, and then studies each components with a resolution matched to its scale. The main feature of the wavelet transform is to hierarchically decompose general functions, as a signal or a process, into a set of approximation functions with different scales.

* Corresponding author. E-mail address: [email protected] (Q. Chen). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.04.025

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Q. Chen, A. Huo / Chaos, Solitons and Fractals 41 (2009) 951–961

Many authors have paid considerable attention to wavelet analysis during the last twenty years. The main advantage of wavelets is their time–frequency localization property. Many signals in areas like music, speech and video image can be efficiently represented by wavelets that are translations and dilations of a single function. Wavelets are a fairly simple mathematical tool with a variety of possible applications. In recent years, multiwavelet theory [16,17] which is an active research field of wavelet analysis, has been applied to many aspects in science and technology, such as, image compress [18], fractals [13], signal processing [19,20], audio coding [21] and so on, mainly because of their ability to offer properties like symmetry, orthogonality, short support at the same time. It is believed that vector-valued wavelets are a class of generalized multiwavelets. Xia and Suter [22] and Chen et al. [22–24] introduced the notion of vector-valued wavelets and studied the existence and construction of orthogonal vector-valued wavelets. Also they investigate the properties of orthogonal vector-valued wavelet packets. Moreover, as reported by Xia, multiwavelets can be generated from the component functions in vector-valued wavelets. Fowler and Li [25] implemented biorthogonal vector-valued wavelet transforms to study fluid flows in oceanography and aerodynamics. Therefore, studying vector-valued wavelets is useful in multiwavelet theory. However, multiwavelets and vector-valued wavelets are different in the following sense. Prefiltering is usually required for discrete multiwavelet transforms [26] but not necessary for discrete vector-valued wavelet transforms. Therefore, it is necessary to study the construction of various vector-valued wavelets. In order to implement the vector-valued wavelet transform, we need to construct various vector-valued wavelets. However, as yet there has not been a general method to obtain biorthogonal vector-valued wavelets. The main objective of this paper is to investigate existence and construction of biorthogonal vector-valued wavelets. Another purpose is to present an algorithm for constructing a class of biorthogonal compactly supported vector-valued wavelet functions. We also discuss the properties of the biorthogonal vector-valued wavelet packets. An outline of the paper is as follows. In Section 2, we describe fundamentals on vector-valued function spaces and introduce vector-valued multiresolution analysis. In Section 3, we give our main result, an algorithm for constructing a pair of biorthogonal compactly supported vector-valued wavelets. In Section 4, two examples will be given. In the final section, we introduce the biorthogonal vector-valued wavelet packets and characterize their properties. 2. Notations and vector-valued multiresolution analysis Throughout this paper, we use the following notations. Let R denote the set of real numbers. Z and Z þ denote all integers and all nonnegative integers, respectively. Define L2 ðRÞn to be the set of all vector-valued functions KðtÞ, i.e., L2 ðRÞn , fKðtÞ ¼ ðk1 ðtÞ; k2 ðtÞ; . . . ; kn ðtÞÞT : ki ðtÞ 2 L2 ðRÞ; i ¼ 1; 2; . . . ; ng, where T means the transpose of a vector, and n 2 Z þ is a constant as well as n P 2. By In and O, we denote the n  n identity matrix and zero matrix, respectively. For arbitrary R KðtÞ 2 L2 ðRÞn , its integration R KðtÞdt is defined to be Z

Z KðtÞdt , R

k1 ðtÞdt;

R

Z

k2 ðtÞdt; . . . ; R

T

Z

kn ðtÞdt

;

ð1Þ

R

and for any KðtÞ ¼ ðk1 ðtÞ; k2 ðtÞ; . . . ; kn ðtÞÞT 2 L2 ðRÞn , define its Fourier transform as follows: Z b KðtÞ  expfictg dt; c 2 R: KðcÞ ,

ð2Þ

For any K 2 L2 ðRÞn ; kKk2 represents the norm of the vector-valued function KðtÞ, i.e., !1=2 n Z X jki ðtÞj2 dt ; kKk2 ,

ð3Þ

R

i¼1

R

which is the norm we use throughout the paper for the vector-valued function space L2 ðRÞn . Let sk K stand for integer translations: sk KðtÞ :¼ Kðt  kÞ; k 2 Z. Set D denote a dilation operator DKðtÞ ¼ 2Kð2tÞ. For two vector-valued functions KðtÞ; CðtÞ 2 L2 ðRÞn , we define their symbol inner product to be Z KðtÞCðtÞ dt; ð4Þ hK; Ci , R

where  means the transpose and the complex conjugate. Definition 1. We say that a pair of vector-valued functions KðtÞ; CðtÞ 2 L2 ðRÞn are biorthogonal, if their integer translations satisfy hKðÞ; Cð  kÞi ¼ d0;k In ;

k 2 Z;

ð5Þ

where d0;k is the Kronecker symbol, that is, as k ¼ 0; d0;k ¼ 1, when k–0; d0;k ¼ 0. Lemma 1. Two vector-valued functions KðtÞ; CðtÞ 2 L2 ðRÞn are biorthogonal if and only if they satisfy X b þ 2mpÞ Cðc b þ 2mpÞ ¼ In ; c 2 R: Kðc m2Z

ð6Þ

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Proof. If KðtÞ and CðtÞ are a pair of biorthogonal vector-valued functions, then we get from (5) that d0;k In ¼ hKðÞ; Cð  kÞi ¼

1 2p

Z R

b CðcÞ b   expfikcg dc ¼ 1 KðcÞ 2p

Z

2p 0

X

b þ 2mpÞ Cðc b þ 2mpÞ  eikc dc; Kðc

m2Z

which implies (6) follows. The converse is obvious. h Definition 2. A sequence of vector-valued functions fKk ðtÞgk2Z  U  L2 ðRÞn is called a Riesz basis of subspace U if (i) for any CðtÞ 2 U, there exists a unique sequence of n  n matrix fM k gk2Z 2 ‘2 ðZÞnn such that X M k Kk ðtÞ; ð7Þ CðtÞ ¼ k2Z

where 2

nn

‘ ðZÞ

8 < ¼ P : Z ! C nn ; kPk2 ¼ :

n X X

!12 jpi;j ðkÞj

i;j¼1 k2Z

2

9 = < þ1 : ;

(ii) there exist constants 0 < C 1 6 C 2 < þ1, such that for any n  n constant matrix sequence fM k gk2Z 2 ‘2 ðZÞnn , it follows that X C 1 kfM k gkI 6 k M k Kk ðtÞk2 6 C 2 kfM k gkI ; k2Z

where kfM k gkI denotes the norm of the matrix sequence fMk gk2Z . For example, we can choose kfM k gkI ¼ where for every k 2 Z; kM k kF denotes Frobenius norm of the matrix M k .

P

2 k2Z kM k kF

12

,

The multiresolution analysis approach is one of the important methods in the construction of wavelets. In what follows, we introduce vector-valued multiresolution analysis and biorthogonal vector-valued wavelets. Definition 3. A vector-valued multiresolution analysis (VMRA) of L2 ðRÞn is a nested sequence of closed subspaces X j of L2 ðRÞn ðj 2 ZÞ, that satisfies the following properties: (i) (ii) (iii) (iv) (v)

X j  X jþ1 8j 2 Z; KðtÞ 2 X 0 () Dl KðtÞ 2 X l 8l 2 Z; KðtÞ 2 X j () sk KðtÞ 2 X j 8j; k 2 Z; S T n 2 j2Z X j ¼ fOg; j2Z X j is dense in L ðRÞ ; hðtÞ : k 2 Zg forms a Riesz basis for X 0 . 9 a vector-valued function  hðtÞ 2 X 0 such that its translates fsk 

Since  hðtÞ 2 X 0  X 1 , by Definition 3 and (7), there exists a finite supported sequence of n  n matrices fAk gk2Z , which has only finite nonzero terms, such that X X hðtÞ ¼  Ak Dsk  hðtÞ ¼ 2  Ak  hð2t  kÞ: ð8Þ k2Z

k2Z

Eq. (8) is called a refinement equation and  hðtÞ is called a vector-valued scaling function. By taking the Fourier transform for ^ ðcÞ is continuous at zero, we have the both sides of (8), and assuming  h ^ðc=2Þ; ^ ðcÞ ¼ Aðc=2Þ h h

c 2 R;

ð9Þ

where AðcÞ ¼

X

Ak  expfikcg:

ð10Þ

k2Z

Let Y j ; j 2 Z, stand for the complementary subspace of X j in X jþ1 and there exist a vector-valued function  ðtÞ 2 L2 ðRÞn such that  j;k ðtÞ ¼ 2j=2  ð2j t  kÞ; j; k 2 Z forms a Riesz basis of X j , i. e., X j ¼ ClosL2 ðRÞn ðSpanf ð2j  kÞ : k 2 ZgÞ;

j 2 Z:

ð11Þ 2

nn

It is clear that  ðtÞ 2 Y 0  X 1 . Hence there exist a finite supported sequences of n  n matrices fBk gk2Z 2 ‘ ðZÞ X X Bk Dsk  hðtÞ ¼ 2  Bk  hð2t  kÞ:  ðtÞ ¼ k2Z

such that ð12Þ

k2Z

Let BðcÞ ¼

X k2Z

Bk  expfikcg:

ð13Þ

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Q. Chen, A. Huo / Chaos, Solitons and Fractals 41 (2009) 951–961

Then, the refinement equation (12) becomes ^ ð2cÞ ¼ BðcÞh ^ðcÞ;

c 2 R:

ð14Þ

e ðtÞ 2 L2 ðRÞn are a pair of biorthogonal vector-valued scaling functions, if there exists another vector-valued We say  hðtÞ;  h e ðtÞ 2 L2 ðRÞn , such that scaling function  h e ð  kÞi ¼ d In ; h hðÞ;  h 0;k

k 2 Z:

ð15Þ

e ðtÞ 2 L2 ðRÞn are called a pair of biorthogonal vector-valued wavelets associated with a pair Vector-valued functions  ðtÞ;  e ðtÞ, if they satisfy of biorthogonal vector-valued scaling functions  hðtÞ and  h e ð  kÞi ¼ O; h hðÞ;  e h h ðÞ;  ð  kÞi ¼ O;

k 2 Z;

ð16Þ

k 2 Z;

e ð  kÞi ¼ d0;k In ; h ðÞ; 

ð17Þ

k 2 Z:

ð18Þ

e ðtÞ also satisfy the following refinement equations: Similar to (8) and (12), e h ðtÞ and   X e e ð2t  kÞ; ek  h  ðtÞ ¼ 2  A h

ð19Þ

k2Z

e ðtÞ ¼ 2  

X

e ð2t  kÞ; ek  h B

ð20Þ

k2Z

e k g; f B e k g 2 ‘2 ðZÞnn are two finitely supported sequences of n  n matrices. Then, we have the following theorems where f A from formulas (8), (15) and (19). e Theorem 1. Suppose that  hðtÞ and h  ðtÞ, defined by (8) and (19), are a pair of biorthogonal vector-valued scaling functions. Then, for any k 2 Z, we have X e mþ2k Þ ¼ 1 d0;k In ; k 2 Z: Am ð A ð21Þ 2 m2Z Proof. By substituting Eqs. (8) and (19) into the biorthogonality relation (15) 8 k 2 Z, we have XXZ XX Z ~ðÞi ¼ 4  ~ð2t  uÞ ð A e u Þ dt ¼ 2  e u Þ d0;k In ¼ h hð  kÞ;  h Am  hð2t  2k  mÞ h Am h  ðt  2k  mÞh ~ðt  uÞ dtð A ¼2

XX m2Z

m2Z

u2Z

R

e u Þ ¼ 2  Am h hð  2k  mÞ; h ~ð  uÞið A

u2Z

XX m2Z

u

m2Z

e u Þ ¼ 2  Am dmþ2k;u ð A

u2Z

X

R

e mþ2k Þ : Am2Z ð A



m2Z

e ðtÞ, defined in (12) and (20), are vector-valued functions in L2 ðRÞn . Then  ðtÞ and  e ðtÞ are a Theorem 2 [23]. Assume  ðtÞ and  pair of biorthogonal vector-valued wavelet functions associated with a pair of biorthogonal vector-valued scaling functions  hðtÞ e ðtÞ, then and  h X

e m Þ ¼ O; Amþ2k ð B

k 2 Z;

ð22Þ

e mþ2k ðBk Þ ¼ O; A

k 2 Z;

ð23Þ

m2Z

X m2Z

X m2Z

e k Þ ¼ 1 d0;k In ; Bmþ2k ð B 2

k 2 Z:

ð24Þ

Thus, both Theorem 2 and formulas (22)–(24) provide an approach to constructing compactly supported biorthogonal vector-valued wavelets. 3. Construction of biorthogonal vector-valued wavelets It is well known that there exists a simple procedure for constructing uniwavelets if an orthogonal uniscaling function is given. For the case of biorthogonal vector-valued wavelets, it seems that a simple method for obtaining them has not been discovered yet. We will proceed to study the construction of biorthogonal vector-valued wavelets and present an algorithm for constructing them. e ðtÞ be a pair of compactly supported biorthogonal vector-valued scaling functions in L2 ðRÞn satisfying Theorem 3. Let  hðtÞ and h  the following equations: h  ðtÞ ¼ 2 

r X k¼0

Ak  hð2t  kÞ;

e ðtÞ ¼ 2   h

r X k¼0

e ð2t  kÞ; ek  h A

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Q. Chen, A. Huo / Chaos, Solitons and Fractals 41 (2009) 951–961

e e ð2tÞ þ e e 2k þ A e 2k2 ¼ A e 2kþ1 þ A e 2k1 ; and set h  y ðtÞ ¼  hð2tÞ þ  hð2t  1Þ; h  y ðtÞ ¼  h h ð2t  1Þ. Suppose A2k þ A2k2 ¼ A2kþ1 þ A2k1 ; A  k ¼ 0; 1; . . . ; dr=2e, where r 2 Z is a constant number and dxe ¼ inffm : x < m 2 Zg. Then e y ðtÞ are a pair of biorthogonal compactly supported vector-valued scaling functions, and supp  h hy ðtÞ  (i) h  y ðtÞ and  e y ðtÞ  ½0; dr=2e. ½0; dr=2e; supp  h e  y ðtÞ satisfy the following refinement equations, respectively, (ii) Vector-valued functions  hy ðtÞ and h hy ðtÞ ¼ 2  

dr=2e X

ðA2k þ A2k2 Þ hy ð2t  kÞ;

ð25Þ

k¼0

e h y ðtÞ ¼ 2  

dr=2e X

e y ð2t  kÞ: e 2k þ A e 2k2 Þ  ðA h

ð26Þ

k¼0

Proof e ðtÞ are a pair of biorthogonal vector-valued scaling functions, then (i) Since  hðtÞ and  h e  y ðt  kÞi ¼ h hð2tÞ þ  hð2t  1Þ; e h ð2t  2kÞ þ e  h ð2t  2k  1Þi  h hy ðtÞ; h e ð2t  2kÞi þ h e ð2t  2k  1Þi þ h e e ð2t  2k  1Þi ¼ h hð2tÞ;  h hð2tÞ;  h hð2t  1Þ; h  ð2t  2kÞi þ h hð2t  1Þ;  h ¼

1 1 1 1 d0;2k In þ d0;2kþ1 In þ d0;2k1 In þ d0;2k In ¼ d0;k In : 2 2 2 2

(ii) From the definition of  hy ðtÞ, we obtain hy ðtÞ ¼ 2  

r X

Ak  hð4t  kÞ þ 2 

k¼0

¼2

dr=2e X

r X

Ak  hð4t  k  2Þ

k¼0

½A2k  hð4t  2kÞ þ A2kþ1 h  ð4t  2k  1Þ þ A2k2  hð4t  2kÞ þ A2k1  hð4t  2k  1Þ

k¼0

¼2

dr=2e X

fðA2k þ A2k2 Þ hð4t  2kÞ þ ðA2kþ1 þ A2k1 Þ hð4t  2k  1Þg

k¼0

¼2

dr=2e X

ðA2k þ A2k2 Þf hð4t  2kÞ þ  hð4t  2k  1Þg ¼ 2 

k¼0

¼2

dr=2e X

dr=2e X

ðA2k þ A2k2 Þf hð2ð2t  kÞÞ þ  hð2ð2t  kÞ  1Þg

k¼0

ðA2k þ A2k2 Þ hy ð2t  kÞ:

k¼0

e e y ðtÞ  ½0; dr=2e and e y ðtÞ ¼ 2  Pdr=2e ð A e 2k þ A e 2k2 Þ h  y ð2t  kÞ. By using (25) and (26), we have supp  h Similarly, we can obtain  h k¼0 e supp  h y ðtÞ  ½0; dr=2e. h Theorem 3 shows a pair of two-scale biorthogonal vector-valued scaling functions with support interval ½0; dre ¼ f0; 1; . . . ; dreg can be changed into those which have support interval ½0; dr=2e. According to Theorem 3, without loss of generality, we only discuss the problems about construction of vector-valued wavelets with 3-coefficient. The main result of the paper is obtained as follows. h e ðtÞ be a pair of 3-coefficient biorthogonal compactly supported vector-valued scaling functions in L2 ðRÞn Theorem 4. Let  hðtÞ and  h satisfying the following equations: hð2tÞ þ 2A1  hð2t  1Þ þ 2A2  hð2t  2Þ;  ðtÞ ¼ 2A0  h e ðtÞ ¼ 2 A e ð2tÞ þ 2 A e ð2t  1Þ þ 2 A e0  e1  e2 e h  h h h ð2t  2Þ:  Assume that there is an integer ‘; 0 6 ‘ 6 2, such that the matrix M below is an invertible matrix:  1 1 e ‘ Þ e ‘ Þ : M2 ¼ A‘ ð; A In  A‘ ð A 2

ð27Þ ð28Þ

ð29Þ

Define 8 Bj ¼ MAj ; j–‘; > > > > < Bj ¼ M1 Aj ; j ¼ ‘; j; ‘ 2 f0; 1; 2g e j ; j–‘; e j ¼ M A > B > > > :e e j ; j ¼ ‘: B j ¼ ðM Þ1 A

ð30Þ

956

Q. Chen, A. Huo / Chaos, Solitons and Fractals 41 (2009) 951–961

Then  ðtÞ ¼ 2B0  hð2tÞ þ 2B1  hð2t  1Þ þ 2B2  hð2t  2Þ; e e e ð2t  2Þ e e e e  ð2tÞ þ 2 B 1  h ð2t  1Þ þ 2 B 2  h  ðtÞ ¼ 2 B 0 h e ðtÞ. are a pair of biorthogonal vector-valued wavelet functions associated with  hðtÞ and  h e0; B e1; B e 2 g satisfy Proof. For convenience, let ‘ ¼ 1. By Theorem 2 and formulas (22)–(24), it suffices to show that fB0 ; B1 ; B2 ; B the following equations: e 2 Þ ¼ O; A0 ð B e 0 Þ ¼ O; A2 ð B e 0 Þ þ A1 ð B e 1 Þ þ A2 ð B e 2 Þ ¼ O; A0 ð B e 0 ðB2 Þ ¼ O; A

ð31Þ ð32Þ ð33Þ ð34Þ

e 2 ðB0 Þ ¼ O; A e 0 ðB0 Þ þ A e 1 ðB1 Þ þ A e 2 ðB2 Þ ¼ O; A e 2 Þ ¼ O; B0 ð B

ð36Þ

e 0 ðB2 Þ ¼ O; B

ð38Þ

e 0 Þ þ B1 ð B e 1 Þ þ B2 ð B e 2 Þ ¼ 1 In : B0 ð B 2 





ð35Þ

ð37Þ

ð39Þ

e0; B e1; B e 2 g are given by (30), then Eqs. (31), (32), (34), (35), (37) and (38) follow from (21). For (33), we get from If fB0 ; B1 ; B2 ; B (21) and (29) as well as (30) that e 0 Þ M  A1 ð A e 1 Þ M 1 þ A2 ð A e 2 Þ M ¼ ½A0 ð A e 0 Þ þ A2 ð A e 2 Þ M  A1 ð A e 1 Þ M 1 e 0 Þ þ A1 ð B e 1 Þ þ A2 ð B e 2 Þ ¼ A0 ð A A0 ð B     1 1 e 1 Þ M  A1 ð A e 1 Þ M 1 ¼ e 1 Þ M 2  A1 ð A e 1 Þ M1 ¼ In  A1 ð A In  A1 ð A 2 2   1 e e ¼ ½A1 ð A 1 Þ  A1 ð A 1 Þ M ¼ O: Similarly, (36) can be established. Finally, we will prove (39) follows: e 0 Þ M þ MA2 ð A e 2 Þ M þ M 1 A1 ð A e 1 Þ M 1 ¼ M½A0 ð A e 0 Þ þ A2 ð A e 2 Þ M þ M 1 A1 ð A e 1 Þ M 1 e 0 Þ þ B1 ð B e 1 Þ þ B2 ð B e 2 Þ ¼ MA0 ð A B0 ð B    1 e 1 Þ M 2 þ A1 ð A e 1 Þ M 1 ¼ M1 ½M 2 A1 ð A e 1 Þ þ A1 ð A e 1 Þ M 1 ¼ M1 M 2 In  A1 ð A 2    e 1 Þ þ M 2 A1 ð A e 1 Þ M 1 ¼ M A1 ð A e 1 Þ þ ðA1 ð A e 1 Þ Þ1 1 In  A1 ð A e 1 Þ A1 ð A e 1 Þ M1 ¼ M½A1 ð A 2   1 1 1 e 1 Þ þ In  A1 ð A e 1 Þ M1 ¼ MIn M 1 ¼ In : ¼ M A1 ð A  2 2 2 Corollary 1. If  hðtÞ defined in (8) is a 3-coefficient orthogonal vector-valued scaling functions and there exists an integer ‘; 0 6 ‘ 6 2, such that the matrix D, defined in the following equation, is not only an invertible but also an Hermitian matrix D2 ¼

 1 1 A‘ A‘ : In  A‘ ðA‘ Þ 2

ð40Þ

Let (

Bj ¼ DAj ;

j–‘;

Bj ¼ D1 Aj ;

j ¼ ‘:

j; ‘ 2 f0; 1; 2g

ð41Þ

Then  ðtÞ, defined in (12), is an orthogonal vector-valued wavelet functions associated with  hðtÞ hð2tÞ þ 2B1  hð2t  1Þ þ 2B2  hð2t  2Þ:  ðtÞ ¼ 2B0 

ð42Þ

Remark. According to the matrix theory, if M is an invertible matrix satisfying M2 ¼ Q , then M is not unique. Hence, matrices M and D defined in (29) and (40), respectively, are not unique. Moreover, in the light of Theorem 4 or Corollary 1, we claim that there exist many distinct vector-valued wavelets associated with a pair of biorthogonal vector-valued scaling functions or orthogonal vector-valued scaling functions.

Q. Chen, A. Huo / Chaos, Solitons and Fractals 41 (2009) 951–961

957

4. Constructive examples e ðtÞ ¼ ½0; 2, be a pair of 3-coefficient biorthogonal vector-valued e ðtÞ 2 L2 ðRÞ2 and supp  hðtÞ ¼ supp  h Example 1. Let  hðtÞ;  h scaling functions satisfying the following equations [17]:  ðtÞ ¼ 2A0  h hð2tÞ þ 2A1  hð2t  1Þ þ 2A2 h  ð2t  2Þ; e e e e ð2t  2Þ; e e e h ðtÞ ¼ 2 A 0   h ð2tÞ þ 2 A 1  h ð2t  1Þ þ 2 A 2  h where A0 ¼ e0 ¼ A

1 4

1 10  15

 12 1 4

A1 ¼

;

5 8  35 64

7  32

pffiffi 2ð1þiÞ 4

! !

2ð1þiÞ 4

;

0

!

pffiffi 1þi 3 8

0

1 4 1 2

A2 ¼

pffiffi ;

1þi 3 8

0 pffiffi

e1 ¼ A

!

0

;

1  10

!

 15 1 4 7 32

e2 ¼ A

; !

 58

:

 35 64

Let ‘ ¼ 0. By using (29) and (30), we get !   1 0 1 0 1 pffiffi ; p ffiffiffi ; M ¼ M¼ 0 77 0 7 0 1 pffiffi ! 1 1  2ð1þiÞ 0 4 10 4 pffiffi pffiffi ; B0 ¼ B1 ¼ @ B2 ¼ pffiffi pffiffi A; 3Þ  147  357 0  7ð1þi 8 0 pffiffi 1 ! 1 5  2ð1þiÞ 0 4 8 4 e e e2 ¼ pffiffi pffiffi ; B1 ¼ @ pffiffi A; B0 ¼ B pffiffi 3Þ  327  5647 0  7ð1þi 8

! 1  10 pffiffi ;  357 ! 1  58 4 pffiffi pffiffi : 7  5647 32

1 4 pffiffi 7 14

From Theorem 4, we conclude that  ðtÞ ¼

2 X

e ðtÞ ¼ 

Bk  hð2t  kÞ;

k¼0

2 X

e ð2t  kÞ ek  h B

k¼0

e ðtÞ. are a pair of biorthogonal vector-valued wavelets associated with  hðtÞ and h  hðtÞ ¼ ½0; 2, be 3-coefficient orthogonal vector-valued scaling functions satisfying: Example 2. Let  hðtÞ 2 L2 ðRÞ2 and supp  hð2tÞ þ 2A1  hð2t  1Þ þ 2A2  hð2t  2Þ; hðtÞ ¼ 2A0   where A0 ¼

0 0

pffiffi ! 2þ 7 8 pffiffi ; 2 7 8

3 8 1 8

A1 ¼

1 8 3 8

pffiffi 2 7 8 pffiffi 2þ 7 8

! ;

A2 ¼

0

!

0

:

Suppose ‘ ¼ 1, By using (40) and (41), we get that pffiffi pffiffi ! pffiffi pffiffi ! 7þ 7 14 pffiffi 7 7 14

D¼ B0 ¼

0 0

7 7 14 pffiffi 7þ 7 14

3 8 1 8

;

D1 ¼ pffiffi 7

!

B1 ¼

;

 2þ8

pffiffi  28 7

1þ 7 2 pffiffi 1 7 2

1 7 2 pffiffi 1þ 7 2 pffiffi !  28 7 pffiffi ;  2þ8 7

; B2 ¼

1 8 3 8

0 0

! :

From Corollary 1, we obtain  ð2tÞ þ 2B1  hð2t  1Þ þ 2B2  hð2t  2Þ  ðtÞ ¼ 2B0 h is orthogonal vector-valued wavelets associated with  hðtÞ. 5. Characterization of biorthogonal vector-valued wavelet packets In order to implement vector-valued wavelet transform, we need to construct biorthogonal vector-valued wavelet packets and investigate their properties. Let us introduce some notations  ðtÞ; F 0 ðtÞ ¼ h ð0Þ Pk

¼ Ak ;

e ðtÞ; e e ðtÞ; F 1 ðtÞ ¼  ðtÞ; e F 0 ðtÞ ¼  h F 1 ðtÞ ¼  ð0Þ ð1Þ e k; P e ¼A e ¼B e k ; k 2 Z: ¼ Bk ; P

ð1Þ Pk

k

k

958

Q. Chen, A. Huo / Chaos, Solitons and Fractals 41 (2009) 951–961

Thus, (8), (12) and (19), (20) become the following equations, respectively, X ðlÞ Pk F 0 ð2t  kÞ; l ¼ 0; 1; F l ðtÞ ¼ 2 

ð43Þ

k2Z

e F l ðtÞ ¼ 2 

X

e ðlÞ e P F 0 ð2t  kÞ; k

l ¼ 0; 1:

ð44Þ

k2Z

For any m 2 Z þ , let s be a unique positive integer such that m ¼ 2s þ l, where l ¼ 0; 1. X ðlÞ Pk F s ð2t  kÞ; l ¼ 0; 1; s 2 Z þ ; F m ðtÞ ¼ F 2sþl ðtÞ ¼ 2 

ð45Þ

k2Z

e F m ðtÞ ¼ e F 2sþl ðtÞ ¼ 2 

X

e ðlÞ e P F s ð2t  kÞ; k

l ¼ 0; 1; s 2 Z þ :

ð46Þ

k2Z

Definition 4. A family of vector-valued functions fF 2sþl ðtÞ : s ¼ 0; 1; 2; . . . ; l ¼ 0; 1g is called vector-valued wavelet packets F 2sþl ðtÞ : s ¼ 0; 1; 2; . . . ; l ¼ 0; 1g is said to be vector-valued with respect to the vector-valued scaling function F 0 ðtÞ, and f e F 0 ðtÞ are a pair of biorthogonal vector-valued scaling functions. wavelet packets with respect to F 0 ðtÞ, where F 0 ðtÞ and e Implementing the Fourier transform for the both sides of (45) and (46) gives b F s ðc=2Þ; c 2 R; l ¼ 0; 1; s 2 Z þ ; F 2sþl ðcÞ ¼ PðlÞ ðc=2Þ b b b ðlÞ e e F s ðc=2Þ; c 2 R; l ¼ 0; 1; s 2 Z þ ; F 2sþl ðcÞ ¼ P ðc=2Þ e

ð47Þ ð48Þ

where PðlÞ ðcÞ ¼

X

e ðlÞ ðcÞ ¼ P

ðlÞ

P k  expfikcg;

k2Z

X

e ðlÞ  expfikcg; P k

c 2 R; l ¼ 0; 1:

k2Z

Therefore, it follows from (6) and (15)–(18) that e ðlÞ ðcÞ þ PðmÞ ðc þ pÞ P e ðlÞ ðc þ pÞ ¼ dm;l In ; PðmÞ ðcÞ P

m; l ¼ 0; 1; c 2 R:

ð49Þ

F s ðtÞ; s ¼ 0; 1; 2; . . .g are a pair of vector-valued wavelet packets with respect to a pair of Lemma 2. If fF s ðtÞ; s ¼ 0; 1; 2; . . .g and f e F 0 ðtÞ, respectively, then, for every s 2 Z þ , it follows that biorthogonal vector-valued scaling functions F 0 ðtÞ and e F s ð  kÞi ¼ dj;k In ; hF s ð  jÞ; e

j; k 2 Z:

ð50Þ

Proof. (i) When s ¼ 0, it follows from (15) that the relation (50) holds. (ii) Suppose s that-(50) follows for the case of 0 6 s < 2 -þ1 -1 where - is a fixed positive integer. Then, when 2 6 s < 2 , we get 2 6 2 < 2 where ½s ¼ maxfm 2 Z; m 6 sg. Order s ¼ 2½s=2 þ q; l ¼ 0; 1. By the induction assumption and Lemma 1, we get that

F ½s=2 ð  kÞi ¼ dj;k In () hF ½s=2 ð  jÞ; e

X

b b F ½s=2 ðc þ 2lpÞ ¼ In : F ½s=2 ðc þ 2lpÞ e

ð51Þ

l2Z

According to (6) and (47)–(49), we obtain X

b b F s ð2c þ 2lpÞ ¼ F s ð2c þ 2lpÞ e

l2Z

X

b e ðlÞ ðc þ lpÞ PðlÞ ðc þ lpÞ b F ½s=2 ðc þ lpÞ e F ½s=2 ðc þ lpÞ P

l2Z

¼

1 X

ðlÞ

P ðc þ rpÞ

r¼0

( X

) b  e b e ðlÞ ðc þ rpÞ F ½s=2 ðc þ rp þ 2jpÞ F ½s=2 ðc þ rp þ 2jpÞ P

j2Z

e ðlÞ ðcÞ þ PðlÞ ðc þ pÞ P e ðlÞ ðc þ pÞ ¼ In : ¼ PðlÞ ðcÞ P By Lemma 1, it follows that (50) is established for the case of 2- 6 k < 2-þ1 . In summary in light of steps (i) and (ii), we complete the proof of the Lemma. h F s ðtÞ; s 2 Z þ g are vector-valued wavelet packets associated with a pair of biorthogonal vectorLemma 3. If fF s ðtÞ; s 2 Z þ g and f e F 0 ðtÞ, respectively, then, for every s ¼ 0; 1; 2; . . ., it follows that valued scaling functions F 0 ðtÞ and e hF 2s ð  kÞ; F 2sþ1 ð  mÞi ¼ O;

k; m 2 Z:

ð52Þ

959

Q. Chen, A. Huo / Chaos, Solitons and Fractals 41 (2009) 951–961

Proof. Based on Parseval equality and Lemma 1 as well as Lemma 2, and formulas (44)–(46), then for every s ¼ 0; 1; 2; . . ., we get that Z Z ^  e 2sþ1 ð  mÞi ¼ 1 e 2sþ1 ðcÞ  expfiðk  mÞcg dc ¼ 1 e b 2s ðcÞ b b 2s ð2cÞ F hF 2s ð  kÞ; F F F F 2sþ1 ð2cÞ  expf2iðk  mÞcg dc 2pZ R p R 1 b e ð1Þ ðcÞ  expf2iðk  mÞcg dc ¼ Pð0Þ ðcÞ b F k ðcÞ e F k ðcÞ P p R Z 1 X ð2lþ1Þp ð0Þ b b e s ðcÞ P e ð1Þ ðcÞ  expf2iðk  mÞcg dc ¼ P ðcÞ F s ðcÞ F p l2Z 2lp ( ) Z X 1 2p ð0Þ  b b e ð1Þ ðcÞ  e2iðmkÞc dc P ðcÞ ¼ F s ðc þ 2lpÞ F s ðc þ 2lpÞ P p 0 l2Z Z p 1 e ð1Þ ðcÞ þ Pð0Þ ðc þ pÞ P e ð1Þ ðc þ pÞ   e2iðmkÞc dc ¼ O: ½Pð0Þ ðcÞ P  ¼ p 0 Theorem 5. If fF s ðtÞ; s ¼ 0; 1; 2; . . .g and f e F s ðtÞ; s ¼ 0; 1; 2; . . .g are vector-valued wavelet packets with respect to a pair of biorF 0 ðtÞ, respectively, then, for any s; r 2 Z þ and k 2 Z, we have thogonal vector-valued scaling functions F 0 ðtÞ and e e ð53Þ hF s ðÞ; F r ð  kÞi ¼ ds;r d0;k In : Proof. For s ¼ r, formula (53) follows by means of Lemma 2. Without loss of generality, we suppose s > r for the case of s–r. Then, rewrite s; r as s ¼ 2½s=2 þ q1 ; r ¼ 2½r=2 þ l1 where q1 ; l1 2 f0; 1g. Case 1. If ½s=2 ¼ ½r=2, then q1 –l1 . From (6) and (47)–(49) it is concluded that (53) holds, i.e., Z Z 1 b b e r ð  kÞi ¼ 1 b b F r ðcÞ  eikc dc ¼ F r ð2cÞ  expf2ikcg dc hF s ðÞ; F F s ðcÞ e F s ð2cÞ e 2pZ R p R 1  e ðl1 Þ  b s ðcÞ b e ¼ Pðq1 Þ ðcÞ F ½2 F ½2r  ðcÞ P ðcÞ  expf2ikcg dc p R Z 1 X ð2lþ1Þp ðq1 Þ b b e ðl1 Þ ðcÞ  expf2ikcg dc ¼ P ðcÞ F ½ s  ðcÞ e F ½ r  ðcÞ P 2 2 p l2Z 2lp ( ) Z X 1 2p ðq1 Þ b  b e e ðl1 Þ ðcÞ  e2ikc dc P ðcÞ ¼ F ½ r  ðc þ 2lpÞ F ½ r  ðc þ 2lpÞ P 2 2 p 0 l2Z Z 1 p ðq1 Þ e ðl1 Þ  e ðl1 Þ ðc þ pÞ e2ikc dc ¼ O: ½P ðcÞ P ðcÞ þ Pðq1 Þ ðc þ pÞ P ¼ p 0 Case 2. For the case of ½s=2–½r=2, we order ½s=2 ¼ 2½½s=2=2 þ q2 ; ½r=2 ¼ 2½½r=2=2 þ l2 ; q2 ; l2 2 f0; 1g. If ½½s=2=2 ¼ ½½r=2=2, then (53) can be established similar to Case 1. Provided that ½½s=2=2–½½r=2=2, then we again order ½½s=2=2 ¼ 2½½½s=2=2=2 þ q3 ; ½½r=2=2 ¼ 2½½½r=2=2=2 þ l3 ; q3 ; l3 2 f0; 1g. Thus, after taking finite steps (denoted by j), we obtain aj ¼ bj ¼ 1;

qj ; lj 2 f0; 1g

ð54Þ

and aj ¼ 1;

bj ¼ 0;

qj ; lj 2 f0; 1g;

ð55Þ

where j

zffl}|ffl{ aj ¼ ½½   ½ s=2   =2;

j

zffl}|ffl{ bj ¼ ½½   ½ r=2   =2:

For the case (54), the result (53) can be established similar to Case 1. For the case (55), we obtain from (16)–(18) that X

b b F 0 ðc þ 2lpÞ ¼ O; F 1 ðc þ 2lpÞ e

c 2 R:

l2Z

Consequently,

Z Z 1 1 b b b e ðl1 Þ ðc=2Þ  expfikcg dc Pðq1 Þ ðc=2Þ b F ½s=2 ðc=2Þ e  F s ðcÞ e F r ðcÞ  expfikcg dc ¼ F ½r=2 ðc=2Þ P 2p ZR 2p R 1 b e ðl2 Þ ðc=4Þ P e ðl1 Þ ðc=2Þ  expfikcg dc ¼    ¼ Pðq1 Þ ðc=2ÞPðq2 Þ ðc=4Þ b F ½s=4 ðc=4Þ e F ½r=4 ðc=4Þ P  2p R" " # # Z Y j j Y 1 b e ðlr Þ ðc=2r Þ  expfikcg dc ¼ Pðqr Þ ðc=2r Þ b F 1 ðc=2j Þ e P F 0 ðc=2j Þ 2p R r¼1 r¼1 !" # " # Z 2jþ1 p Y j j  c  Y   X c   1 b ðqr Þ c ðlr Þ c e b e ¼  eikc dc P P F 1 j þ 2lp F 0 j þ 2lp 2p 0 2r 2 2 2r r¼1 r¼1 l2Z " # # Z 2jþ1 p "Y j j Y 1 e ðlr Þ ðc=2r Þ  expfikcg dc ¼ O: ¼ Pðqr Þ ðc=2r Þ  O  P 2p 0 r¼1 r¼1

F r ð  kÞi ¼ hF s ðÞ; e

Therefore, for any r; s 2 Z þ and k 2 Z, the result (53) follows. h,

960

Q. Chen, A. Huo / Chaos, Solitons and Fractals 41 (2009) 951–961

e j and Y j ; Y e j by virtue of a series of subspaces of In what follows, we will investigate the decomposition of subspaces X j ; X vector-valued wavelet packets. Furthermore, the direct decomposition relation for space L2 ðRÞn is presented. Let us introduce a notation DX ¼ fDKðtÞ : KðtÞ 2 Xg, where X  L2 ðRÞn . For any s 2 Z þ , define X Pk F s ðt  kÞ; fP k g 2 ‘2 ðZÞnn g; ð56Þ Xs ¼ fKðtÞ : KðtÞ ¼ k2Z

e s ¼ f KðtÞ e e X : KðtÞ ¼

X

e k g 2 ‘2 ðZÞnn g: ek e P F s ðt  kÞ; f P

ð57Þ

k2Z

Then, we have the following relations X0 ¼ X 0 ; X1 ¼ Y 0 . Lemma 4 [27]. 8s 2 Z þ , the space DXs can be decomposed into the direct sum of spaces X2sþl ; l 2 f0; 1g, i.e., DXs ¼ X2s e 2s U X e 2sþ1 ; where U denote the direct sum. es ¼ X and D X

U

X2sþ1 ,

For U; V  R, let aU ¼ fau : u 2 U; a 2 Rg; U þ V ¼ fu þ v : u 2 U; v 2 Vg; U  V ¼ fu  v : u 2 U; v 2 Vg. For a fixed positive P i e e integer m, denote by e Em ¼ m i¼0 2 f0; 1g; Em ¼ E m  E m1 . Theorem 6. The set of vector-valued functions fF s ðt  kÞ; s 2 Em ; k 2 Zg forms a Riesz basis of Dm Y 0 . In particular, the collection fF s ð  kÞ; s 2 Z þ ; k 2 Zg constitutes a Riesz basis of L2 ðRÞn . U Proof. By Lemma 4, we obtain that DX0 ¼ X0 X1 , i.e., DX 0 ¼ X 0  Y 0 . It can inductively be proved by using Lemma 4 that ] ] Xi ; and Dm X 0  Dm Y 0 ¼ Dmþ1 X 0 ; that is Dm Y 0 ¼ Xr : Dm X 0 ¼ r2Em i2e Em Therefore, the aggregate fF s ð  kÞ; s 2 Em ; k 2 Zg forms a Riesz basis of Dm Y 0 . Moreover ! !! ] ] ] m ] ] ] ¼ D Y 0 ¼ X0 Xr Xr : L2 ðRÞn ¼ X 0 06m

06m

r2Em

r2Z þ

That is to say, the set fF s ð  kÞ; s 2 Z þ ; k 2 Zg constitutes a Riesz basis of the space L2 ðRÞn . h e 0. Corollary 2. The set of vector-valued functions f e F s ðt  kÞ; s 2 Em ; k 2 Zg forms a Riesz basis of Dm Y Corollary 3. For every m 2 N, the family of vector-valued functions fF s ð2j t  kÞ; k 2 Z; j 2 Z; s 2 Em g constitutes a Riesz basis of space L2 ðRÞn . Proof. Since the family fF s ðt  kÞ; k 2 Z; s 2 Em g forms a Riesz basis of Dm Y 0 , then for every j 2 Z, the sequence fF s ð2j t  kÞ; k 2 Z; s 2 Em g constitutes a Riesz basis of subspace Dj Dm Y 0 ¼ Dmþj Y 0 . Consequently, for every m 2 N, we have ] j m ] mþj ] j D D Y0 ¼ D Y0 ¼ D Y 0: ð58Þ j2Z

j2Z

j2Z

Therefore, fF a ð2j t  kÞ; k; j 2 Z; s 2 Em g constitutes a Riesz basis of space L2 ðRÞn . h 6. Conclusion The notion of vector-valued multiresolution analysis is introduced and the construction of biorthogonal vector-valued wavelets is investigated. An algorithm for constructing a class of biorthogonal vector-valued compactly supported wavelets associated with a pair of biorthogonal vector-valued compactly supported scaling functions has been proposed. This method is suitable for the case of orthogonal vector-valued compactly supported wavelets. The notion of biorthogonal vector-valued wavelet packets is introduced and their properties are characterized by virtue of matrix theory and functional analysis method. Three biorthogonality formulas concerning the wavelet packets are established. The direct decomposition relation for space L2 ðRÞn is discussed. Relation to some physical theories such as E-infinity Cantorian space–time theory is also considered. Acknowledgements The authors would like to thank the anonymous referees for their comments. The authors also thank Prof. M.S. El Naschie and Dr. H.G. Boehm for their help. This work is supported by the Natural Science Foundation of China (10571113) and Xi’an University of Architecture and technology(JC0718, RC0701).

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961

References [1] El Naschie MS. A guide to the mathematics of E-infinity Cantorian space–time theory. Chaos, Solitons & Fractals 2005;25(5):955–64. [2] Iovane G, Giordano P. Wavelet and multiresolution analysis: nature of e1 Cantorian space–time. Chaos, Solitons & Fractals 2007;32(3):896–910. [3] El Naschie MS. Remarks on super strings, fractal geometry, Nagasawa’s diffusion and Cantorian space–time. Chaos, Solitons & Fractals 1997;8(11):1873–86. [4] El Naschie MS. Feigenbaum scenario for turbulence and Cantorian E-infinity theory of high energy particle physics. Chaos, Solitons & Fractals 2007;32(3):911–5. [5] Cones A. Non-commutative geometry. New York: Academic Press; 1994. [6] El Naschie MS. On 336 kissing spheres in 10 dimensions, 528 P-Brane states in 11 dimensions and the 60 elementary particles of the standard model. Chaos, Solitons & Fractals 2005;24(2):447–57. [7] Perkins D. Introduction to high energy physics. Cambridge; 2000. [8] Kaku M. Introduction to superstrings and M-theory. New York: Springer; 1999. [9] El Naschie MS. Determining the number of Higgs particles starting from general relativityand various other field theories. Chaos, Solitons & Fractals 2005;23(3). [10] El Naschie MS. Hilbert space, the number of Higgs particles and the quantum two-slit experiment. Chaos, Solitons & Fractals 2006;27(1):9–13. [11] El Naschie MS. A review of E-infinity and the quantum two-slit experiment. Chaos, Solitons & Fractals 2004;19(1):209–36. [12] Iovane G, Laserra E, Tortoriello FS. Stochastic self-similar and fractal universe. Chaos, Solitons & Fractals 2004;20(2):415–26. [13] Iovane G, Giordano P. Wavelet and multiresolution analysis: nature of e1 Cantorian space–time. Chaos, Solitons & Fractals 2007;32(3):896–910. [14] Iovane G. Mohamed El Naschie’s ð1Þ Cartorian space–time and its consequences in cosmology. Chaos, Solitons & Fractals 2005;25(3):775–9. [15] Svozil K. Computational universes. Chaos, Solitons & Fractals 2005;25(4):845–59. [16] Chui CK, Lian J. A study of orthonormal multiwavelets. Appl Numer Math 1996;20(1):273–98. [17] Yang S, Cheng Z, Wang H. Construction of biorthogonal multiwavelets. Math Anal Appl 2002;276(1):1–12. [18] Zhang N, Wu X. Lossless of color masaic images. IEEE Trans Image Delivery 2006;15(6):1379–88. [19] Efromovich S, lakey J, Pereyia M, Tymes N. Data-diven and optimal denoising of a signal and recovery of its derivation using multiwavelets. IEEE Trans Signal Process 2004;52(3):628–35. [20] Telesca L, lapenna V, Alexis V. Multiresolution wavelet analysis of earthquakes. Chaos, Solitons & Fractals 2004;22(3):741–8. [21] Li W. Vector transform and image coding. IEEE Trans Circuits Syst Video Technol 1991;1:297–307. [22] Xia XG, Suter BW. Vector-valued wavelets and vector filter banks. IEEE Trans Signal Process 1996;44(3):508–18. [23] Chen Q, Cheng Z, Wang C. Existence and construction of compactly supported biorthogonal multiple vector-valued wavelets. J Appl Math Comput 2006;22(3):101–15. [24] Chen Q, Cheng Z. A study on compactly supported orthogonal vector-valued wavelets and wavelet packets. Chaos, Solitons & Fractals 2007;31(4):1024–34. [25] Fowler JE, Li H. Wavelet transforms for vector fields using omnidirectionally balanced multiwavelets. IEEE Trans Signal Process 2002;50(12):3018–27. [26] Xia XG, Geronimo JS, Hardin DP, Suter BW. Design of prefilters for discrete multiwavelet transforms. IEEE Trans Signal Process 1996;44(1):25–35. [27] Chen Q, Shi Z. Biorthogonal multiple vector-valued multivariate wavelet packets associated with a dilation matrix. Chaos, Solitons & Fractals 2008;35(2):323–32.