The orthogonal interpolating balanced multiwavelet with rational coefficients

Chaos, Solitons and Fractals 41 (2009) 892–899

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Chaos, Solitons and Fractals

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The orthogonal interpolating balanced multiwavelet withrational coefficients

Rui Li a, Guochang Wu b,*

a Institute of Applied Mathematics, College of Mathematics and Information Science, Henan University, Kaifeng 475001, PR Chinab School of Science, Xi’an Jiaotong University, P.O. Box 1860, Xi’an 710049, PR China

a r t i c l e i n f o

Article history:Accepted 22 April 2008

Communicated by Prof. Ji-Huan He

0960-0779/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.chaos.2008.04.019

* Corresponding author.E-mail address: [email protected] (G. Wu)

a b s t r a c t

In this paper, we construct the orthogonal interpolating multiwavelet of multiplicity r ¼ 4with the balancing property of order 1 and with rational coefficients. At first, we introducethe notations of multiwavelet, interpolating and balancing. Secondly, for an orthogonalmultiwavelet of multiplicity r ¼ 4 having totally interpolating property, we deduce thatthe corresponding filter of the orthogonal multiscaling function with totally interpolatingproperty has the parametric expression. Then, similarly to r ¼ 2, we prove that there noexists any interpolating orthogonal multiwavelet with the symmetry. At last, we constructsome examples of the orthogonal multiwavelets of multiplicity r ¼ 4 with totally interpo-lating property and the balancing property of order 1 by the parametric way.

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1. Introduction

Looking at nature it becomes obvious that it is not continuous, not periodic. However it is self-similar. Mohamed ElNaschie’s E-infinity theory [1–6] has introduced a mathematical formulation to describe phenomena that is resolutiondependent. E-infinity appears to be clearly a new framework for understanding and describing nature. In detail, eð1Þ

space–time is an infinite dimensional fractal, that happens to have D ¼ 4 as the expectation value for topological dimension.And the topological value 3 + 1 means that in our low energy resolution, the world appears to us as if it were four-dimen-sional. Consequently, it all depends on the energy scale through which we are making our observation. The best candidate forthis purpose is the wavelet transform. It is actually a simple mathematical tool that cuts up data or functions into differentfrequency components, and then investigates each components with a resolution matched to its scale. Mathematicians andengineers have paid considerable attention to the wavelet transform. Indeed, engineers have discovered that it can beapplied in all environments where the signal analysis is used. Wavelets are mathematical functions that cut up data into dif-ferent frequency components, then people can study each component with a resolution matched to its scale. Therefore,wavelets and multiresolution analysis are good mathematical tools to support EI Naschie’s picture of the resolution depen-dence of the observations.

In recent years, this idea has been extended to the so-called multiwavelet case [7–13], in which the signal is expressed as alinear combination of dilates and translates of several functions. Since there are several functions, there is more freedom in thedesign of multiwavelets than scalar wavelets, and as a result, multiwavelets can simultaneously possess many desired prop-erties such as short support, orthogonality, symmetry, and vanishing moments, which a single wavelet cannot possess simul-taneously. This suggests that multiwavelet systems can provide perfect reconstruction, good performance at the boundaries(symmetry), and high approximation order (vanishing moments). Although they look more attractive in theory than scalarwavelets, multiwavelets have yet to realize their advantages in practical applications. One reason for this might be due to

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R. Li, G. Wu / Chaos, Solitons and Fractals 41 (2009) 892–899 893

the difficulties involved in their proper discrete implementation [11]. The implementation of a multiwavelet transform differsfrom that of a scalar wavelet transform in that the filter bank requires several input streams. That is, a discrete multiwavelettransform employs a multiple-input, multipleoutput (MIMO) filterbank. To generate the multiple streams, some kind of pre-filtering of the signal must be performed before the implementation of the multifilter bank. However, there are often manyways to do such prefiltering, and different prefilters may lead to different performances. In other words, the analysis of a signalbased on the multiwavelet transform depends not only on the signal and the multifilter bank, but also on the prefilter. Thisinconveniences the user and blurs the foreground of multiwavelet applications. It would be far more convenient if the mul-tiwavelet system could be designed so that the projection coefficients for a given scale were the uniform samples of the signalwithout prefiltering, that is, if a multiwavelet sampling theorem were to hold. Scalar systems admitting such a theorem areknown to exist [17]. Unfortunately, the Haar system is the only two-band orthogonal compactly supported system that pos-sesses this property. (High approximation order M-band systems are available.) In an attempt to obtain two-band interpola-tory multiwavelet systems with high approximation order, Lebrun and Vetterli designed balanced multiwavelets [16].

Sampling theorems play a basic role in digital signal processing. They ensure that continuous signals can be representedand processed by their discrete samples. The classical Shannon Sampling Theorem asserts that bandlimited signals can beexactly represented by their uniform samples as long as the sampling rate is not less than the Nyquist rate. This theoremhas been proved to be fundamental in many applications of signal processing and communication theory. When we usethe wavelet decompositions in digital signal processing, the coefficients in the high level representation can be chosen tobe samples of the continuous signal. However, for the wavelet decomposition, the Mallat algorithm is often used, and FIRfilters are preferred, although filters with exponentially decaying impulse responses also provide satisfactory results formany applications. Because it was shown by Xia and Zhang [17] that compactly supported orthogonal scaling functions can-not be cardinal (except the Haar wavelet), Selesnick [14] studied interpolating orthogonal multiwavelets and refinable func-tions and obtained sampling theorem in the multiwavelet subspace.Then, Zhang [15] characterized all totally interpolatingbiorthogonal finite impulse response (FIR) multifilter banks of multiplicity two, and provided a design framework for corre-sponding compactly supported multiwavelet systems with high approximation order.

In this paper, we will construct the orthogonal interpolating multiwavelets of multiplicity r ¼ 4 with the balancing prop-erty of order 1 and with rational coefficients. At first, we introduce the notations of multiwavelet,interpolating and balanc-ing. Secondly, for an orthogonal multiwavelet of multiplicity r ¼ 4 having totally interpolating property, we give that thecorresponding filter of the orthogonal multiscaling function with totally interpolating property has the parametric expres-sion. Then, similarly to r ¼ 2, we prove that there no exists any interpolating orthogonal multiwavelet functions with thesymmetry. At last, we construct some examples of the orthogonal multiwavelets of multiplicity r ¼ 4 with totally interpo-lating property and the balancing property of order 1 by the parametric way.

This paper is organized as follows. Section 2 is of a preliminary character: it contains various results and some notations.In Section 3, we construct the orthogonal interpolating multiwavelet of multiplicity r ¼ 4 with the balancing property of or-der 1 and with rational coefficient. Then, we give corresponding examples to prove our theory.

2. Main notations

In the multiwavelet system, multiresolution analysis produced by the scalar scaling function is generalized to the case ofthe multiscaling functions. Thus, we have more flexibility. In the same time, the corresponding multiwavelet basis is madeof the translations and the dilations of multiwavelet functions. In the following, we firstly give the definitions of themultiscaling functions and the multiwavelet functions.

Definition 1. In the multiwavelet system of multiplicity r, let uiðxÞ; i ¼ 0;1; . . . ; r � 1 be the multiwavelet functions and/iðxÞ; i ¼ 0;1; . . . ; r � 1 be the multiscaling functions. And let

WðxÞ ¼ ½u0ðxÞ;u1ðxÞ; . . . ;ur�1ðxÞ�T; UðxÞ ¼ ½/0ðxÞ;/1ðxÞ; . . . ;/r�1ðxÞ�

T;

similarly to the scalar wavelet, UðxÞ;WðxÞ satisfy two scaling equations:

UðxÞ ¼ 2X

k

HkUð2x� kÞ ð1Þ

WðxÞ ¼ 2X

k

GkUð2x� kÞ ð2Þ

where Hk;Gk are r � r coefficient matrices, k 2 Z. Let HðzÞ ¼P

kHkz�k, GðzÞ ¼P

kGkz�k, where z ¼ ejw. Then, HðzÞ and GðzÞ arenamed as the filters associated to the multiwavelet system, i.e. the multiwavelet filterbanks.

For the orthogonal multiwavelet system UðxÞ;WðxÞ, it is obvious that they must satisfy:

h/iðx� kÞ;/jðx� lÞi ¼ di;jdk;l; huiðx� kÞ;ujðx� lÞi ¼ di;jdk;l; h/iðx� kÞ;ujðx� lÞi ¼ 0; k; l 2 Z:

Define the modulation matrix of the multiwavelet filter banks:

HmðzÞ ¼HðzÞ Hð�zÞGðzÞ Gð�zÞ

� �:

894 R. Li, G. Wu / Chaos, Solitons and Fractals 41 (2009) 892–899

Thus, by the orthogonal condition, we have:

HmðzÞHTmðz�1Þ ¼ I2r : ð3Þ

For single wavelet transforms, the discrete implementation automatically follows from their multiresolution structure,i.e., tree-structured twochannel filterbanks. In the tree-structured filterbank, lowpass and highpass filters are explicitly used,which is tight with the lowpass and the bandpass properties of the scaling and wavelet functions, respectively. Although, formultiwavelet transforms, the discrete implementation also follows from their multiresolution structure, the tree-structuredfilterbank becomes a tree-structured vector filterbank. For a tree-structured vector filterbank, the lowpass and the highpassproperties for the two vector filters are not as clear as those for the two filters in single wavelet transforms. In order to have areasonable decomposition for discrete multiwavelet transforms, prefiltering is necessary. In [11], balanced multiwaveletswere studied, where prefiltering for these kinds of multiwavelets is not necessary, but other properties, such as the shortsupportness and the smoothness, are not as good as the GHM.

Defining the band-Toeplitz matrix corresponding to the lowpass analysis filter:

L ¼

� � � � � � � � � � � � � � � � � �� H1 H2 � � � Hr�1 � � �� H�1 H0 � � � Hr�3 � � ��

26664

37775:

Then, the matrix scaling equation and wavelet decomposition can be parameterized [8].In the following, we will give other definitions used in our paper.

Definition 2. The orthogonal multiwavelet system is called having the balancing property of order 1, if the lowpasssynthesis operator LT preserves U ¼ ½. . . ;1;1;1; . . . �T. That is:

LTU ¼ U:

Furthermore, define

½m0ðzÞ;m1ðzÞ; . . . ;mr�1ðzÞ�T ¼ HðzrÞ½1; z�1; . . . ; z1�r �T: ð4Þ

Thus, we have

Lemma 1. The orthogonal multiwavelet system has the balancing property of order 1 [13] if and only if the polynomialz1�2r þ z2�2r þ � � � þ 1 can be divided exactly by the term m0ðzÞ þm1ðzÞ þ � � � þmr�1ðzÞ.

Then, we give the definition of totally interpolating property in an orthogonal multiwavelet system:

Definition 3 [15]. A vector function FðxÞ ¼ ½f0ðxÞ; f1ðxÞ; . . . ; fr�1ðxÞ�T is said to be an interpolating multifunction if FðtÞ satisfiesthe condition:

FðnÞ; F nþ 1r

� �; . . . ; F nþ r � 1

r

� �� ¼ dðnÞIr :

If the multiscaling function and the multiwavelet function UðxÞ;WðxÞ in an orthogonal multiwavelet system has the inter-polating property, we say that the system is totally interpolating.

3. Main results

In this paper, we restrict our focus to the case of the orthogonal multiwavelet system of multiplicity r ¼ 4, i.e.,UðxÞ ¼ ½/0ðxÞ;/1ðxÞ;/2ðxÞ;/3ðxÞ�

T. If UðxÞ has totally interpolating property, according to Definition 3, we have,

UðnÞ;U nþ 14

� �;U nþ 2

4

� �;U nþ 3

4

� �� ¼ dðnÞI4;

i.e.,

/0ð0Þ ¼ 1; /0n4

� �¼ 0; n 2 Z; n–0; /1

14

� �¼ 1; /1

n4

� �¼ 0; n 2 Z; n–1:

/224

� �¼ 1; /2

n4

� �¼ 0; n 2 Z; n–2 � /3

34

� �¼ 1; /3

n4

� �¼ 0; n 2 Z; n–3:

Due to the matrix scaling Eq. (1), for x ¼ 0, we have

½/0ð0Þ;/1ð0Þ;/2ð0Þ;/3ð0Þ�T ¼ 2H0½/0ð0Þ;/1ð0Þ;/2ð0Þ;/3ð0Þ�

T;

then, for the matrix H0, we have a11 ¼ 12 ; a21 ¼ a31 ¼ a41 ¼ 0.


When x ¼ 14, we have

/014

� �;/1

14

� �;/2

14

� �;/3

14

� �� T

¼ 2H0 /012

� �;/1

12

� �;/2

12

� �;/3

12

� �� T

;

thus, for the matrix H0, we have a23 ¼ 12 ; a13 ¼ a33 ¼ a43 ¼ 0.

Therefore, we get

H0 ¼

1=2 a12 0 a14

0 a22 1=2 a24

0 a32 0 a34

0 a42 0 a44

26664

37775:

In the same way, we can get other matrices Hk, then, in the case of r ¼ 4, the interpolating orthogonal filter of multiscalingfunctions with totally interpolating property has the parametrization express:

HðzÞ ¼

1=2 h1ðzÞ 0 h5ðzÞ0 h2ðzÞ 1=2 h6ðzÞ

z�1=2 h3ðzÞ 0 h7ðzÞ0 h4ðzÞ z�1=2 h8ðzÞ

26664

37775: ð5Þ

In the same way, if WðxÞ has totally interpolating property, according to the matrix scaling Eq. (2), the orthogonal filter ofmultiwavelet functions with totally interpolating property has the parametrization express:

GðzÞ ¼

1=2 g1ðzÞ 0 g5ðzÞ0 g2ðzÞ 1=2 g6ðzÞ

z�1=2 g3ðzÞ 0 g7ðzÞ0 g4ðzÞ z�1=2 g8ðzÞ

26664

37775: ð6Þ

We know, the symmetry of multiwavelet system plays an important role in some application, such as mapping process-ing. Thus, it is important to construct multiwavelet functions and multifilter bank with the symmetry. However, we will tes-tify, similarly to r ¼ 2, that there no exists any interpolating orthogonal multiwavelet functions with the symmetry [12].

Lemma 2 [16]. Matrix filter HðwÞ is called symmetric, if the following relations hold:

HðwÞdiagð�e�2jT0w; . . . ;�e�2jTr�1wÞ ¼ diagð�e�4jT0w; . . . ;�e�4jTr�1wÞHð�wÞ;

where, Ti is the symmetric origin of the function /i, ‘‘+” implies that /i is symmetric, ‘‘�” implies /i is antisymmetric, HðzÞ ¼ HðejwÞ.

Theorem 1. If the orthogonal multiscaling filter HðzÞ satisfies Eq. (5), Then, there no exists any interpolating orthogonal multiscal-ing filter with the symmetry.

Proof. Suppose the conclusion holds, from Lemma 1, we have

HðwÞdiagð�e�2jT0w;�e�2jT1w;�e�2jT2w;�e�2jT3wÞ ¼ diagð�e�4jT0w;�e�4jT1w;�e�4jT2w;�e�4jT3wÞHð�wÞ:

Furthermore, we have

T0 ¼ 0; T1 ¼14; T2 ¼

12; T3 ¼

34;

that is

h1ð�wÞ ¼ e�12jwh1ðwÞ; h2ð�wÞ ¼ e

12jwh2ðwÞ; h3ð�wÞ ¼ e


52jwh4ðwÞ;

h5ð�wÞ ¼ e�32jwh5ðwÞ; h6ð�wÞ ¼ e�



32jwh8ðwÞ:

Therefore, when w ¼ p, because hiðwÞ; ði ¼ 1;2; . . . ;8Þ are 2p-periodic, we get:

h1ðpÞ ¼ �jh1ðpÞ; h2ðpÞ ¼ jh2ðpÞ; h3ðpÞ ¼ �jh3ðpÞ; h4ðpÞ ¼ jh4ðpÞ;h5ðpÞ ¼ jh5ðpÞ; h6ðpÞ ¼ �jh6ðpÞ; h7ðpÞ ¼ jh3ðpÞ; h8ðpÞ ¼ �jh8ðpÞ;

i.e.,

h1ðpÞ ¼ h2ðpÞ ¼ h3ðpÞ ¼ h4ðpÞ ¼ h5ðpÞ ¼ h6ðpÞ ¼ h7ðpÞ ¼ h8ðpÞ ¼ 0: ð7Þ

Also, when w ¼ p, we get

h1ð2pÞ ¼ h2ð2pÞ ¼ h3ð2pÞ ¼ h4ð2pÞ ¼ h5ð2pÞ ¼ h6ð2pÞ ¼ h7ð2pÞ ¼ h8ð2pÞ ¼ 0: ð8Þ


Due to orthogonality condition HðwÞHð�wÞT þ Hðwþ pÞHð�wþ pÞT ¼ I4, we have

h1ðwÞh1ð�wÞ þ h1ðwþ pÞh1ð�wþ pÞ þ h5ðwÞh5ð�wÞ þ h5ðwþ pÞh5ð�wþ pÞ ¼ 12:

Because hiðwÞ; ði ¼ 1;2; . . . ;8Þ are 2p-periodic, for w ¼ 0, we have

h1ð0Þh1ð0Þ þ h1ðpÞh1ðpÞ þ h5ð0Þh5ð0Þ þ h5ðpÞh5ðpÞ ¼12;

this contradicts Eqs. (7) and (8).Therefore, we complete the proof. h

In the following, we will construct the orthogonal multiwavelet of multiplicity r ¼ 4 with totally interpolating propertyand the balancing property of order 1.

According to Eq. (4), we have

½m0ðzÞ;m1ðzÞ;m2ðzÞ;m3ðzÞ�T ¼ Hðz4Þ½1; z�1; z�2; z�3�T:

Due to Eq. (5), we get

m0ðzÞ ¼12þ h1ðz4Þz�1 þ h5ðz4Þz�3; m1ðzÞ ¼

12

z�2 þ h2ðz4Þz�1 þ h6ðz4Þz�3;

m2ðzÞ ¼12

z�4 þ h3ðz4Þz�1 þ h7ðz4Þz�3; m3ðzÞ ¼12

z�6 þ h4ðz4Þz�1 þ h8ðz4Þz�3:

Therefore, we have

m0ðzÞ þm1ðzÞ þm2ðzÞ þm3ðzÞ ¼12

z�6 þ 12

z�4 þ 12

z�2 þ 12þ ½h1ðz4Þ þ h2ðz4Þ þ h3ðz4Þ þ h4ðz4Þ�z�1 þ ½h5ðz4Þ þ h6ðz4Þ

þ h7ðz4Þ þ h8ðz4Þ�z�3:

Because the multiwavelet system satisfies the balancing property of order 1, from Lemma 1, we have that the polynomialz�7 þ z�6 þ � � � þ 1 can be divided exactly by the term m0ðzÞ þm1ðzÞ þm2ðzÞ þm3ðzÞ. Thus,

½h1ðz4Þ þ h2ðz4Þ þ h3ðz4Þ þ h4ðz4Þ�z�1 þ ½h5ðz4Þ þ h6ðz4Þ þ h7ðz4Þ þ h8ðz4Þ�z�3 ¼ 12

z�7 þ 12

z�5 þ 12

z�3 þ 12

z�1;

that is,

h1ðz4Þ þ h2ðz4Þ þ h3ðz4Þ þ h4ðz4Þ ¼ 12þ 1

2z�4;h5ðz4Þ þ h6ðz4Þ þ h7ðz4Þ þ h8ðz4Þ ¼ 1

2þ 1

2z�4:

In fact, we have proved:

Theorem 2. Let UðxÞ;WðxÞ satisfy totally interpolating property, i.e., Eqs. (5) and (6) hold. Then, if the multiwavelet systemsatisfies the balancing property of order 1, we have


2z�4; ð9Þ


2z�4: ð10Þ

Suppose UðxÞ satisfies self orthogonality property: HðzÞHTðz�1Þ þ Hð�zÞHTð�z�1Þ ¼ I4. According to HðzÞ;GðzÞ satisfy Eqs.(5) and (6), we can give a concrete way to construct GðzÞ, such that it satisfies Eq. (3).

For the sake of the simplification, let AðzÞ;BðzÞ;CðzÞ;DðzÞ; EðzÞ; FðzÞ;MðzÞ;NðzÞ:

AðzÞ ¼h1ðzÞ h1ð�zÞh2ðzÞ h2ð�zÞ

� �; BðzÞ ¼

h3ðzÞ h3ð�zÞh4ðzÞ h4ð�zÞ

� �; CðzÞ ¼

h5ðzÞ h5ð�zÞh6ðzÞ h6ð�zÞ

� �;

DðzÞ ¼h7ðzÞ h7ð�zÞh8ðzÞ h8ð�zÞ

� �; EðzÞ ¼

g1ðzÞ g1ð�zÞg2ðzÞ g2ð�zÞ

� �; FðzÞ ¼


� �;

MðzÞ ¼g5ðzÞ g5ð�zÞg6ðzÞ g6ð�zÞ

� �; NðzÞ ¼


� �:

Due to Eqs. (3) and (5), when HðzÞHTðz�1Þ þ Hð�zÞHTð�z�1Þ ¼ I4 holds, we have

AðzÞATðz�1Þ þ CðzÞCTðz�1Þ ¼ 12

I2;

BðzÞBTðz�1Þ þ DðzÞDTðz�1Þ ¼ 12

I2;

AðzÞBTðz�1Þ þ CðzÞDTðz�1Þ ¼ 02:

ð11Þ


From the orthogonality condition GðzÞGTðz�1Þ þ Gð�zÞGTð�z�1Þ ¼ I4, and due to Eq. (6), we have

EðzÞETðz�1Þ þMðzÞMTðz�1Þ ¼ 12

I2;

FðzÞFTðz�1Þ þ NðzÞNTðz�1Þ ¼ 12

I2;

EðzÞFTðz�1Þ þMðzÞNTðz�1Þ ¼ 02:

ð12Þ

In the same way, if HðzÞGTðz�1Þ þ Hð�zÞGTð�z�1Þ ¼ 04 holds, according to Eqs. (5) and (6), we deduce

AðzÞETðz�1Þ þ CðzÞMTðz�1Þ ¼ �12

I2; BðzÞFTðz�1Þ þ DðzÞNTðz�1Þ ¼ �12

I2;

AðzÞFTðz�1Þ þ CðzÞNTðz�1Þ ¼ 02; BðzÞETðz�1Þ þ DðzÞMTðz�1Þ ¼ 02:

ð13Þ

Let
EðZÞ ¼ �AðZÞ; FðZÞ ¼ �BðZÞ; MðZÞ ¼ �CðZÞ; NðZÞ ¼ �DðZÞ:
If Eq. (11) holds, we get Eqs. (12) and (13). Therefore, from above discuss, we get:

Theorem 3. Suppose that the corresponding filterbank HðzÞ;GðzÞ of a multiwavelet system UðxÞ;WðxÞ satisfies Eqs. (5) and (6).Then, when UðxÞ satisfies self orthogonality property, the necessary condition that functions UðxÞ;WðxÞ form an orthogonalmultiwavelet system is:

g1ðzÞ ¼ �h1ðzÞ; g2ðzÞ ¼ �h2ðzÞ; g3ðzÞ ¼ �h3ðzÞ; g4ðzÞ ¼ �h4ðzÞ;g5ðzÞ ¼ �h5ðzÞ; g6ðzÞ ¼ �h6ðzÞ; g7ðzÞ ¼ �h7ðzÞ; g8ðzÞ ¼ �h8ðzÞ:

In the following, we will construct some examples of the orthogonal multiwavelets of multiplicity r ¼ 4 with totally inter-polating property and the balancing property of order 1.

The concrete steps follow: According to the balancing conditions (9) and (10) and self orthogonality condition (11), bysolving the Eq. (5), we can get concrete parametric values. Thus, we can obtain lowpass filters Hk. At last, according to The-orem 3, we can get highpass filters Gk.

Example 1. We will construct an orthogonal interpolating balanced multiwavelet with minimal compact support. Let

h1ðzÞ ¼ a0 þ a1z�1; h2ðzÞ ¼ a4z�1 þ a5z�2; h3ðzÞ ¼ a8 þ a9z�1; h4ðzÞ ¼ a12z�1 þ a13z�2;

h5ðzÞ ¼ a2 þ a3z�1; h6ðzÞ ¼ a6z�1 þ a7z�2; h7ðzÞ ¼ a10 þ a11z�1; h8ðzÞ ¼ a14z�1 þ a15z�2:

In order to get the orthogonal multiwavelet with the balancing property of order 1 satisfying Eqs. (9) and (10), we need tosolve the following equations:

a0 þ a8 ¼12; a1 þ a4 þ a9 þ a12 ¼

12; a5 þ a13 ¼ 0; ð14Þ

a2 þ a10 ¼12; a3 þ a6 þ a11 þ a14 ¼

12; a7 þ a15 ¼ 0: ð15Þ

In order to get orthogonality condition HðzÞHTðz�1Þ þ Hð�zÞHTð�z�1Þ ¼ I4, we need to solve the following equations:

h1ðzÞh1ðz�1Þ þ h1ð�zÞh1ð�z�1Þ þ h5ðzÞh5ðz�1Þ þ h5ð�zÞh5ð�z�1Þ ¼ 12: ð16Þ

That is, a20 þ a2

1 þ a22 þ a2

3 ¼ 14.

Thus, we can solve the equation bank which consists of all equalities satisfying self orthogonality conditions and Eqs.(14)–(16):

h1ðzÞ ¼14þ 1

4z�1; h2ðzÞ ¼

14

z�1 þ 14

z�2; h3ðzÞ ¼14� 1

4z�1; h4ðzÞ ¼

14

z�1 � 14

z�2;

h5ðzÞ ¼14� 1

4z�1; h6ðzÞ ¼

14

z�1 � 14

z�2; h7ðzÞ ¼14þ 1

4z�1; h8ðzÞ ¼

14

z�1 þ 14

z�2:

At last, we obtain the filter Hk:

H0 ¼

12

14 0 1

4

0 0 12 0

0 14 0 1

4

0 0 0 0

26664

37775; H1 ¼

0 14 0 � 1

4

0 0 14

14

12 � 1

4 0 14

0 14

12

14

26664

37775; H2 ¼

0 0 0 00 1

4 0 � 14

0 0 0 00 � 1

4 0 14

26664

37775:

According to Theorem 3, we get an orthogonal multiwavelet system UðxÞ;WðxÞ such that



Example 2. We will consider the case of filter Hk with longer compact support.

Let

h1ðzÞ ¼ a1 þ a2z�2 þ a3z�3 þ a4z�4 þ a5z�5; h2ðzÞ ¼ a11z�1 þ a12z�2 þ a13z�3 þ a14z�4 þ a15z�5;



h7ðzÞ ¼ a26 þ a27z�2 þ a28z�3 þ a29z�4 þ a30z�5; h8ðzÞ ¼ a36z�1 þ a37z�2 þ a38z�3 þ a39z�4 þ a40z�5:

In order to get the orthogonal multiwavelet with the balancing property of order 1 satisfying Eqs. (9) and (10), we need tosolve the following equations:

a1 þ a21 ¼ 12

a11 þ a31 ¼ 12

a2 þ a12 þ a22 þ a32 ¼ 0

a3 þ a13 þ a23 þ a33 ¼ 0

a4 þ a14 þ a24 þ a34 ¼ 0

a5 þ a15 þ a25 þ a35 ¼ 0

8>>>>>>>>><>>>>>>>>>:

;

a6 þ a26 ¼ 12

a16 þ a36 ¼ 12

a7 þ a17 þ a27 þ a37 ¼ 0

a8 þ a18 þ a28 þ a38 ¼ 0

a9 þ a19 þ a29 þ a39 ¼ 0

a10 þ a20 þ a30 þ a40 ¼ 0

8>>>>>>>>><>>>>>>>>>:

; ð17Þ

In order to get orthogonality condition HðzÞHTðz�1Þ þ Hð�zÞHTð�z�1Þ ¼ I4, we need to solve the following equations:

h1ðzÞh1ðz�1Þ þ h1ð�zÞh1ð�z�1Þ þ h5ðzÞh5ðz�1Þ þ h5ð�zÞh5ð�z�1Þ ¼ 12:

That is,

a21 þ a2

2 þ a23 þ a2

4 þ a25 þ a2

6 þ a27 þ a2

8 þ a29 þ a2

10 ¼ 14

a1a2 þ a2a4 þ a3a5 þ a6a7 þ a7a9 þ a8a10 ¼ 0a1a4 þ a6a9 ¼ 0

8><>: ð18Þ

Thus, we can solve the equation bank which consists of all equalities satisfying self orthogonality conditions and Eqs. (17)and (18):

h1ðzÞ ¼14þ 1

8z�2 þ 1

8z�3 � 1

8z�4 þ 1

8z�5; h2ðzÞ ¼

14

z�1 þ 18

z�2 þ 18

z�3 þ 18

z�4 � 18

z�5;

h3ðzÞ ¼14� 1

8z�2 � 1

8z�3 þ 1

8z�4 � 1

8z�5; h4ðzÞ ¼

14

z�1 � 18

z�2 � 18

z�3 � 18

z�4 þ 18

z�5;

h5ðzÞ ¼14� 1

8z�2 � 1

8z�3 þ 1

8z�4 � 1

8z�5; h6ðzÞ ¼

14

z�1 � 18

z�2 � 18

z�3 � 18

z�4 þ 18

z�5;

h7ðzÞ ¼14þ 1

8z�2 þ 1

8z�3 � 1

8z�4 þ 1

8z�5; h8ðzÞ ¼

14

z�1 þ 18

z�2 þ 18

z�3 þ 18

z�4 � 18

z�5:

Thus, we obtain the filter Hk:

H0 ¼

12

14 0 1

4

0 0 12 0

0 14 0 1

4

0 0 0 0

26664

37775; H1 ¼

0 0 0 00 1

4 0 14

12 0 0 00 1

412

14

26664

37775; H2 ¼

0 18 0 � 1

8

0 18 0 � 1

8

0 � 18 0 1

8

0 � 18 0 1

8

26664

37775;

H3 ¼

0 18 0 � 1

8

0 18 0 � 1

8

0 � 18 0 1

8

0 � 18 0 1

8

26664

37775; H4 ¼

0 � 18 0 1

8

0 18 0 � 1

8

0 18 0 � 1

8

0 � 18 0 1

8

26664

37775; H5 ¼

0 18 0 � 1

8

0 � 18 0 1

8

0 � 18 0 1

8

0 18 0 � 1

8

26664

37775:

According to Theorem 3, we get an orthogonal multiwavelet system UðxÞ;WðxÞ such that


Of course, we can construct longer filters with rational coefficient Hk.

4. Conclusion

Nature shows itself as an arena where the laws of physics appear at each scale in a self-similar way, linked to the reso-lution of the observations which has been introduced by El Naschie’s E-infinity theory. In this paper, inspired by these theory,we construct the orthogonal interpolating multiwavelets of multiplicity r ¼ 4 with the balancing property of order 1 and


with rational coefficients. For an orthogonal multiwavelet of multiplicity r ¼ 4 having totally interpolating property, we givethat the corresponding filter of orthogonal multiscaling functions with totally interpolating property has the parametricexpression. Then, similarly to r ¼ 2, we prove that there no exists any interpolating orthogonal multiwavelet with the sym-metry. At last, we construct some examples of the orthogonal multiwavelets of multiplicity r ¼ 4 with totally interpolatingproperty and the balancing property of order 1 by the parametric way.

Acknowledgements

This work was supported by Innovation Scientists and Technicians Troop Construction Projects of Henan Province (No.084100510012), the Natural Science Foundation of Henan Province of China (No. 0611053200) and the Natural ScienceFoundation for the Education Department of Henan Province of China (No. 2006110001). The authors would like to thankProf. Ji-Huan He for his many valuable suggestions that lead to a significant improvement of our manuscript.

References

[1] El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19(1):209–36.[2] El Naschie MS. A guide to the mathematics of E-infinity Cantorian spacetime theory. Chaos, Solitons & Fractals 2005;25(5):955–64.[3] El Naschie MS. Hilbert space, the number of Higgs particles and the quantum two-slit experiment. Chaos, Solitons & Fractals 2006;27(1):9–13.[4] El Naschie MS. Hilbert, Fock and Cantorian spaces in the quantum two-slit Gedanken experiment. Chaos, Solitons & Fractals 2006;27(1):39–42.[5] El Naschie MS. Intermediate prerequisites for E-infinity theory. Chaos, Solitons & Fractals 2006;30:622–8.[6] Iovane G, Giordano P. Wavelets and multiresolution analysis: nature of eð1Þ Cantorian space–time. Chaos, Solitons & Fractals 2007;32:896–910.[7] Goodman TN, Lee SL. Wavelets of multiplicity r. Trans Am Math Soc 1994;342(3):307–24.[8] Chui CK, Lian J. A study of orthogonal multi-wavelets. Appl Numer Math 1996;20(3):273–98.[9] Cui LH, Cheng ZX. An algorithm for symmetric orthogonal multiwavelets by matrix symmetric extension. Appl Math Comput 2004;149(1):227–43.

[10] Lebrun J, Vetterli M. Balanced multiwavelets theory and design. IEEE Trans Signal Process 1998;46(4):1119–24.[11] Xia XG, Hardin DP, Geronimo J. Design of prefilters for discrete multiwavelets transforms. IEEE Trans Signal Process 1996;44(2):25–35.[12] Zhou DX. Interpolatory orthogonal multiwavelets and refinable functions. IEEE Trans Signal Process 2002;50(3):520–7.[13] Selesnick IW. Multiwavelet bases with extra approximation properties. IEEE Trans Signal Process 1998;46(11):2898–909.[14] Selesnick IW. Interpolating multiwavelet bases and the sampling theorem. IEEE Trans Signal Process 1999;47(6):1615–21.[15] Zhang JK. Design of interpolating biorthogonal multiwavelet systems with compact support. Appl Comput Harmon Anal 2001;11:420–38.[16] Plonka G, Strela V. Construction of multiscaling functions with approximation and symmetry. SIAM J Math Anal 1998;29:450–81.[17] Xia XG, Zhang Z. On sampling theorem, wavelet and wavelet transforms. IEEE Trans Signal Process 1993;41:3524–35.

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The orthogonal interpolating balanced multiwavelet with rational coefficients