Chaos, Solitons and Fractals 42 (2009) 2420–2424

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals

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

Generalized orthogonal multiwavelet packets

Lei Sun a,*, Gang Li b

a School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, Chinab Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China

a r t i c l e i n f o a b s t r a c t

Article history:Accepted 30 March 2009

0960-0779/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.chaos.2009.03.111

* Corresponding author.E-mail address: sunlei97@163.com (L. Sun).

In this paper, the definition, construction and properties of generalized orthogonal multi-wavelet packets are given based on orthogonal multiwavelet packets. By the choices ofmultifilter banks satisfying perfect reconstruction conditions, we can construct a lot of gen-eralized orthogonal multiwavelet packets and generalize the known results.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

It is well known that nature is clearly not continuous, not periodic, but self-similar. Mohamed EI Naschie has introduced amathematical formulation to describe phenomena, that is resolution dependent. His E-infinity view appears to be clearly anew framework for understanding and describing nature. As reported in [1–3], space-time is an infinite dimensional fractalthat happens to have D ¼ 4 as the expectation value for topological dimension. The topological value 3þ 1 means that, in ourlow energy resolution, the world appears to us if it were four-dimensional. In [4–7], Iovane shows that the dimensionchanges if we consider different energies, corresponding to different lengths-scale in universe. The Fourier’s transform isa mathematical tool to consider the motion either in the frequency domain or in the time domain. It cannot be simulta-neously described in the frequency domain as well as in the time domain. Because of this, we need a transform that takessimultaneously account into the two aspects. Fortunately, the wavelet transform, which permits a multiresolution analysisof data with different behavior on different scales, can make up for the disadvantage.

Wavelet packets, due to their good characteristics, have attracted much attention recently. It is well known that theorthogonal basis generated by a wavelet of L2ðRÞ has poor frequency localization. To overcome this disadvantage, Wickerha-user [8] introduced the concept of orthogonal wavelet packets. Chui [9] extended it to non-orthogonal wavelet packet, mak-ing spline wavelet suitable to wavelet packets. Z.X. Cheng [10] introduced wavelet packets with scaling matrix further andgeneralized some results in [8,9]. However, all these results on wavelet packets are constricted to uni-wavelet. Advances onmultiwavelet packets are due to Yang [11], Chen [12–14], Hang [15] and so on. In this paper, we give the definition, construc-tion and properties of generalized orthogonal multiwavelet packets based on the results in [11] and get more general results.

2. Multiresolution analysis of L2ðRÞ

We begin with some basic theory and notations to be used throughout this paper. Let Z and R be the set of all integers andreal numbers, respectively. For a vector-valued function UðxÞ ¼ f/1ðxÞ;/2ðxÞ; . . . ;/rðxÞg

T (/iðxÞ 2 L2ðRÞ;1 6 i 6 r), define aclosed subspace Vj � L2ðRÞ by

Vj ¼ ClosL2ðRÞðspanfUðajx� kÞgÞ; k 2 Z:

The multiwavelet construction is associated with multiresolution analysis (MRA). More precisely, a MRA of multiplicity r is anested sequence of the closed subspaces fVjgj2Z in L2ðRÞ satisfying the following conditions:

. All rights reserved.

L. Sun, G. Li / Chaos, Solitons and Fractals 42 (2009) 2420–2424 2421

(1) � � � � V�1 � V0 � V1 � � � � ;(2) \j2ZVj ¼ f0g; [j2ZVj is dense in L2ðRÞ;(3) f ðxÞ 2 Vj if and only if f ðaxÞ 2 Vjþ1; j 2 Z;(4) there exists a vector-valued function UðxÞ ¼ f/1ðxÞ;/2ðxÞ; . . . ;/rðxÞg

T 2 V0 such that the sequence fUðx� kÞ; k 2 Zg isan orthogonal basis of V0.

The function UðxÞ is called an orthogonal multiscaling function of multiplicity r and the MRA is generated by UðxÞ. FromUðxÞ 2 V0 # V1, there exists an r � r matrix sequence fPkgk2Z which is called matrix low-pass multifilter such that

UðxÞ ¼Xk2Z

PkUðax� kÞ: ð1Þ

Let W0 be the orthogonal complement of V0 to V1, i.e., V1 ¼ V0 �W0. If there exists a vector-valued functionWðxÞ ¼ fw1ðxÞ;w2ðxÞ; . . . ;wða�1ÞrðxÞg

T such that fWðx� kÞ; k 2 Zg forms an orthogonal basis for W0. We call WðxÞ an orthogonalmultiwavelet associated with UðxÞ. From WðxÞ 2W0 # V1, there exists an ða� 1Þr � r matrix sequence fQkgk2Z which is calledmatrix high-pass multifilter such that

WðxÞ ¼Xk2Z

Q kUðax� kÞ: ð2Þ

Set

PðxÞ ¼ 1aXk2Z

Pk � e�ikx and QðxÞ ¼ 1aXk2Z

Q k � e�ikx:

By taking the Fourier transform of two sides of (1) and (2), we have

bUðxÞ ¼ Pxa

� �bU xa

� �; x 2 R; ð3Þ

and

bWðxÞ ¼ Qxa

� �bU xa

� �; x 2 R ð4Þ

We refer PðxÞ;QðxÞ as to the refinement mask and wavelet mask, respectively.

Lemma 2.1. Let UðxÞ ¼ f/1ðxÞ;/2ðxÞ; . . . ;/rðxÞgT be a multiscaling function. Then UðxÞ is orthogonal if and only if

Xk2Z

bUðxþ 2kpÞbU�ðxþ 2kpÞ ¼ Ir; x 2 R;

where the superscript � denotes the conjugate transpose of matrix.

Lemma 2.2. Let UðxÞ ¼ f/1ðxÞ;/2ðxÞ; . . . ;/rðxÞgT be an orthogonal multiscaling function satisfying (1) and PðxÞ be the refine-

ment mask. Then

Xa�1

k¼0

Pxþ 2kp

a

� �P�

xþ 2kpa

� �¼ Ir ; x 2 R: ð5Þ

Let WðxÞ ¼ fw1ðxÞ;w2ðxÞ; . . . ;wða�1ÞrðxÞgT be an orthogonal multiwavelet corresponding to WðxÞ and QðxÞ be the wavelet

mask. Then

Xa�1

k¼0

Pxþ 2kp

a

� �Q �

xþ 2kpa

� �¼ Or�ða�1Þr ; x 2 R; ð6Þ

Xa�1

k¼0

Qxþ 2kp

a

� �Q �

xþ 2kpa

� �¼ Iða�1Þr ; x 2 R: ð7Þ

Obviously, the formulae (5)–(7) are equivalent to

Xk2Z

PkPTkþai ¼ ad0;iIr; ð8ÞX

k2Z

PkQTkþai ¼ Or�ða�1Þr ; ð9Þ

and X

k2Z

Q kQTkþai ¼ ad0;iIða�1Þ�r ð10Þ

respectively. The multifilter banks fPk; Qkgk2Z are said to satisfy perfect reconstruction (P.R.) conditions if fPk; Q kgk2Z satisfy(8)–(10).

2422 L. Sun, G. Li / Chaos, Solitons and Fractals 42 (2009) 2420–2424

3. Generalized orthogonal multiwavelet packets

In this section, we give the definition, construction and properties of generalized orthogonal multiwavelet packets. LetUðxÞ ¼ f/1ðxÞ;/2ðxÞ; . . . ;/rðxÞg

T be an orthogonal multiscaling function with the corresponding matrix low-pass filtersequence fP0

kgk2Z . Suppose that WðxÞ ¼ fw1ðxÞ;w2ðxÞ; . . . ;wða�1ÞrðxÞgT is the corresponding orthogonal multiwavelet whose

matrix high-pass filter sequence is fQkgk2Z . We divide the ða� 1Þr dimension vector-valued function WðxÞ into r dimensionvector-valued function as follows:

W1ðxÞ ¼ fw1ðxÞ;w2ðxÞ: . . . ;wrðxÞg;W2ðxÞ ¼ fwrþ1ðxÞ;wrþ2ðxÞ: . . . ;w2rðxÞg;

..

.

Wa�1ðxÞ ¼ fwða�2Þrþ1ðxÞ;wða�2Þrþ2ðxÞ; . . . ;wða�1ÞrðxÞg:

Let Pik be a high-pass filter sequence associated to WiðxÞ; 1 6 i 6 a� 1. Denote

U0ðxÞ ¼ UðxÞUiðxÞ ¼ WiðxÞ; 1 6 i 6 a� 1:

(ð11Þ

Then the two scaling equations can be written as

U0ðxÞ ¼Pk2Z

P0kU0ðax� kÞ

UiðxÞ ¼Pk2Z

PikU0ðax� kÞ; 1 6 i 6 a� 1:

8><>: ð12Þ

Orthogonal multiwavelet packets with respect to UðxÞ are defined by

UalþiðxÞ ¼Xk2Z

PikUlðax� kÞ; 1 6 i 6 a� 1; ð13Þ

where l ¼ 0;1;2; . . . ; i ¼ 0;1; . . . ;a� 1. One can consult this in [11] and we call them traditional orthogonal multiwaveletpackets.

Now we take another multifilter banks fG0k ;Hkgk2Z satisfying P.R. conditions, where fG0

kgk2Z is an r � r matrix low-pass fil-ter sequence, and fHkgk2Z is an ða� 1Þr � r matrix high-pass filter sequence. For any k 2 Z, we suppose that

Hk ¼

hk11 hk

12 . . . hk1r

hk21 hk

22 . . . hk2r

..

. ... ..

. ...

hkða�1Þ1 hk

ða�1Þ2 . . . hkða�1Þr

2666664

3777775:

Denote

G1k ¼

hk11 hk

12 . . . hk1r

hk21 hk

22 . . . hk2r

..

. ... ..

. ...

hkr1 hk

r2 . . . hkrr

2666664

3777775

G2k ¼

hkðrþ1Þ1 hk

ðrþ1Þ2 . . . hkðrþ1Þr

hkðrþ2Þ1 hk

ðrþ2Þ2 . . . hkðrþ2Þr

..

. ... ..

. ...

hkð2rÞ1 hk

ð2rÞ2 . . . hkð2rÞr

26666664

37777775;

..

.

Gða�1Þk ¼

hkðða�2Þrþ1Þ1 hk

ða�2Þrþ1Þ2 . . . hkða�2Þrþ1Þr

hkðða�2Þrþ2Þ1 hk

ðða�2Þrþ2Þ2 . . . hkðða�2Þrþ2Þr

..

. ... ..

. ...

hkðða�1Þrþ1Þ1 hk

ðða�1Þrþ1Þ2 . . . hkðða�1Þrþ1Þr

26666664

37777775:

L. Sun, G. Li / Chaos, Solitons and Fractals 42 (2009) 2420–2424 2423

We replace Pik in (13) with Gi

k for 0 6 i 6 a� 1 and have

U00ðxÞ ¼Pk2Z

G0kU0ðax� kÞ

U0iðxÞ ¼Pk2Z

GikU0ðax� kÞ; 1 6 i 6 a� 1:

8><>:

Define

U0alþiðxÞ ¼Xk2Z

GikU0lðax� kÞ; 1 6 i 6 a� 1; ð14Þ

where l ¼ 0;1;2; . . . ; i ¼ 0;1; . . . ;a� 1. We call U0alþiðxÞ generalized orthogonal multiwavelet packets with respect to UðxÞ.In what follows, we give the properties of generalized orthogonal multiwavelet packets U0alþiðxÞ.For n ¼ 0;1; . . ., in order to describe the Fourier transform of U0alþiðxÞ, we need write n in the form of

n ¼X1j¼1

�jaj�1; �j 2 f0;1;2; . . . ;a� 1g: ð15Þ

Theorem 3.1. Let n be written in the form of (15). Then the Fourier transform of generalized orthogonal multiwavelet packet is

Y1 x

bU 0nðxÞ ¼j¼1

G�j ðe�iaj ÞbU 00ð0Þ: ð16Þ

Proof. For arbitrary non-negative integer n, we show it by induction. If n ¼ 0, clearly, (16) holds. We suppose that fors0 s0 s0þ1

0 6 n 6 a (16) holds. Now consider a < n < a . Noting that

n ¼ ana

h iþ �1 ¼ an1 þ �1;

where n1 ¼ ½na� ¼Ps0

j¼1�jþ1aj�1 and as0�16 n1 6 as0 . By induction, we have

bU 0n1ðxÞ ¼

Y1j¼1

G�jþ1xa

� �e�ix

aj bU 00ð0Þ:

From bU 0nðxÞ ¼ G�1 x

a

� �bU 0n1

xa

� �, so

bU 0nðxÞ ¼Y1j¼1

G�j e�ixaj bU 00ð0Þ;

which implies that, for as0 < n < as0þ1, (16) holds, as required. h

Theorem 3.2

hU0nðx� jÞ;U0nðx� kÞi ¼ dj;kIr ; where j; k 2 Z: ð17Þ

Proof. For arbitrary non-negative integer n, we show it by induction. If n ¼ 0, clearly, (17) holds. Suppose that 0 6 n 6 as0 ,(17) holds. We now consider as0 < n < as0þ1. Denote n ¼ a½na� þ �1 ¼ an1 þ �1�1 2 f0;1; . . . ;a� 1g, and have

hU0nðx� jÞ;U0nðx� kÞi ¼ 12p

Z 1

�1

bU 0nðxÞbU 0�n ðxÞeiðk�jÞdx

¼ 12p

Z 1

�1G�1

xa

� �bU 0n1

xa

� �bU 0�n1

xa

� �G�1 � x

a

� �eiðk�jÞdx

¼ 12p

Xþ1l¼�1

Z 2apðlþ1Þ

2aplG�1

xa

� �bU 0n1

xa

� �bU 0�n1

xa

� �G�1 � x

a

� �eiðk�jÞdx

¼ 12p

Z 2ap

0G�1

xa

� � Xþ1l¼�1

bU 0n1

xaþ 2pl

� �bU 0�n1

xaþ 2pl

� �( )G�1 � x

a

� �eiðk�jÞdx:

Noting that as0�16 n1 6 as0 , we have

hU0n1ðx� jÞ;U0n1

ðx� kÞi ¼ dj;kIr ;

that is,

Xþ1l¼�1

bU 0n1

xaþ 2pl

� �bU 0�n1

xaþ 2pl

� �¼ Ir :

2424 L. Sun, G. Li / Chaos, Solitons and Fractals 42 (2009) 2420–2424

Consequently,

hU0nðx� jÞ;U0nðx� kÞi ¼ 12p

Z 2ap

0G�1

xa

� �IrG

�1 � xa

� �dx ¼ 1

2p

Z 2p

0

Xa�1

l¼0

G�1xþ 2pi

a

� �IrG

�1 � xþ 2pia

� �eiðk�jÞdx ¼ dj;kIr;

which implies that, for as0 < n < as0þ1, (17) holds and the conclusion follows. h

Theorem 3.3

hU0anðx� jÞ;U0anðx� kÞi ¼ O; where j; k 2 Z and i 2 f1;2; . . . ;a� 1g: ð18Þ

Proof. From Theorem 3.2, for n P 0, it follows that

hU0nðx� jÞ;U0nðx� kÞi ¼ dj;kIr ;

that is,

Xþ1l¼�1

bU 0n xaþ 2pl

� �bU 0�n xaþ 2pl

� �¼ Ir :

So

hU0anðx� jÞ;U0anþiðx� kÞi ¼ 12p

Z 1

�1

bU 0anðxÞbU 0�anþiðxÞeiðk�jÞdx

¼ 12p

Xþ1l¼�1

Z 2apðlþ1Þ

2aplG0 x

a

� �bU 0n xa

� �bU 0�n xa

� �Gi� x

a

� �eiðk�jÞdx

¼ 12p

Z 2ap

0G0 x

a

� � Xþ1l¼�1

bU 0n xaþ 2pl

� �bU 0�n xaþ 2pl

� �( )Gi� x

a

� �eiðk�jÞdx

¼ 12p

Z 2ap

0G0 x

a

� �IrG

i� xa

� �eiðk�jÞdx

¼ 12p

Z 2p

0

Xa�1

i¼0

G0 xþ 2pia

� �Gi� xþ 2pi

a

� �eiðk�jÞdx ¼ 0: �

Having the above properties of U0nðxÞðn ¼ 0;1; . . .Þ, we can decompose L2ðRÞ and omit it here.

4. Conclusion

In this paper, we give the definition, construction and properties of generalized orthogonal multiwavelet packets. Due tomany choices of the multifilter banks fG0

k ;Hkgk2Z satisfying P.R. conditions, we can get many generalized orthogonal multi-wavelet packets. So the case considered in [11] is a special one of this paper.

References

[1] EI Naschie MS. A guide to the mathematics of E-Infinity Cantorian spacetime theory. Chaos, Solitons & Fractals 2005;25(5):955–64.[2] EI Naschie MS. Hilbert, Fock and Cantorian spaces in the quantum two-slit gedanken experiment. Chaos, Solitons & Fractals 2006;27(1):39–42.[3] EI Naschie MS. Hilbert space, the number of Higgs particles and the quantum two-slit experiment. Chaos, Solitons & Fractals 2006;27(1):9–13.[4] Iovane G, Laserra E, Tortoiello FS. Stochastic self-semilar and fractal nuiverse. Chaos, Solitons & Fractals 2004;20(2):415–26.[5] Iovane G. Waveguiding and mirroring effects in stochastic self-similar and fractal universe. Chaos, Solitons & Fractals 2004;23(3):691–700.[6] Iovane G, Mohamed EI. Naschie’s e1 Cantorian space-time and its consequences in cosmology. Chaos, Solitons & Fractals 2005;25(3):775–9.[7] Iovane G, Giordano P. Wavelets and multiresolution analysis: nature of e1 Cantorian space-time. Chaos, Solitons & Fractals 2007;32(3):896–910.[8] Wickerhauser MV. A coustic signal compression with wavelet packets in wavelets. A tutorial in theory and applications, Academic Boston

1992:679–700.[9] Chui CK, Chun L. Non-orthonarmal wavelet packets. SIAM J Math Anal 1993;24:712–38.

[10] Zhengxing Cheng. Wavelet packets with scaling matrix. Chinese J Eng Math 1994;1:15–28.[11] Shouzhi Yang, Zhengxing Cheng. Orthogonal multiwavelet packets with scale a. Mathematica Spplicata 2000;13:61–5.[12] Qingjiang Chen, Zhi Shi. Construction and properties of orthogonal matrix-valued wavelets and wavelet packets. Chaos, Solitons & Fractals

2008;37(1):75–86.[13] Qingjiang Chen, Zhi Shi. Biorthogonal multiple vector-valued multivariate wavelet packets associated with a dilation matrix. Chaos, Solitons & Fractals

2008;35(2):323–32.[14] Qingjiang Chen, Huaixin Cao, Zhi Shi. Construction and characterizations of orthogonal vector-valued multivariate wavelet packets. Chaos, Solitons &

Fractals 2008;35(2):323–32.[15] Jincang Han, Zhengxing Cheng, Qingjiang Chen. A study of biorthogonal multiple vector-valued wavelets. Chaos, Solitons & Fractals

2009;40(4):1574–87.