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JOURNAL OF MATHEMATICAL PHYSICS VOLUME 39, NUMBER 4 APRIL 1998

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Wavelets and quantum algebrasA. LuduDepartment of Physics and Astronomy, Louisiana State University,Baton Rouge, Louisiana 70803-4001

M. GreinerInstitut fur Theoretische Physik, Technische Universita¨t, D-01062 Dresden, Germany

J. P. DraayerDepartment of Physics and Astronomy, Louisiana State University,Baton Rouge, Louisiana 70803-4001

~Received 23 October 1995; accepted for publication 12 November 1997!

Wavelets, known to be useful in nonlinear multiscale processes and in multireso-lution analysis, are shown to have aq-deformed algebraic structure. The translationand dilation operators associate to any scaling equation a nonlinear, two parameteralgebra. This structure can be mapped onto the quantum group slq(2) in one limit,and approaches a Fourier series generating algebra, in another limit. A dualitybetween any scaling function and its corresponding nonlinear algebra is obtained.Examples for the Haar andB-wavelets are worked out in detail. ©1998 Ameri-can Institute of Physics.@S0022-2488~98!00703-8#

I. INTRODUCTION

Interest in wavelet theory and its applications in the multiresolution analysis1 has grown overthe last decade, involving new and linking disparate fields of research from pure mathematiphysics to down-to-earth signal engineering. It is being widely applied in signal processindata compression,2–4 pattern recognition,5 statistical physics and turbulence,6 jet dynamics,7 fieldtheory,8,9 solid state physics,10–12quantum mechanical applications,13–16nonlinear dynamics,6,7,17

soliton theory,18,19 etc.The rise in interest in wavelet theory is motivated by the fact that Fourier analysis is

ineffective when dealing with nonlinear models and localize sharp features.6–18 A central propertyof wavelet theory is the ability to expand and analyze functions with respect to a self-similof localized basis functions~scaling functions and wavelets! and to processes them locally, without affecting the scale. The wavelet method is recursive and therefore ideal for computaapplications. Moreover, the scaling functions and the corresponding wavelets are very wellized both in the time and frequency domains. Wavelets are the natural tool in the analyphenomena where different space/time scales occur. They provide mathematical represethat can handle both analytical and numerical difficulties due to singular phenomena.6,7 LikeFourier analysis, wavelet theory uses expansion functions with different characteristic sHowever, Fourier analysis has the advantage of being based on a simple and solid alfoundation.20 A challenge is to see if an algebraic framework for the foundation of the wavanalysis can be constructed.

This paper is a first step in this direction. It is mainly concerned in obtaining a cloalgebraic method for the construction of the scaling and wavelet functions. We present resuthe Haar andB-wavelets. The purpose of this paper is to show that wavelets are not only mresolution bases of self-similar functions but they have a definite nonlinear symmetry assowith a q-deformation of the Fourier series generating algebra. In a following second paper wshow that wavelets also have variational properties: all multiscale equations follow from Hton’s equation of an infinite-dimensional Hamiltonian system.

We also show that the central object of wavelet theory, the two-scale equation, has anbraic counterpart, arising from a quantum algebra.21 Applications of this result can be realizeusing supercomputers and efficient numerical schemes which have led to discrete modecomplex continuous systems or of genuine discrete physical systems defined on lattice

23460022-2488/98/39(4)/2346/16/$15.00 © 1998 American Institute of Physics

1 Jun 2006 to 130.39.180.124. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp

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2347J. Math. Phys., Vol. 39, No. 4, April 1998 Ludu, Greiner, and Draayer

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discrete symmetries related to complete solvability22 or to discrete-continuous transitions,23 andtheir basic tool, the finite difference equations, are new and central items in any analydiscrete systems. On uniform lattices the symmetry algebras of partial differential equationsunchanged by the discretization.24 In the case of nonuniform lattices one needs to introdgeneralized symmetries, like quantum algebras.25 This is an example when theq-deformation ofsome initial symmetries can play an important role. An outcome of the present paperextension of the traditional Fourier analysis towards wavelet expansions that can be accomby q-deforming the Fourier algebra.

The finite-difference and scaling operators involved in the dilation equation are algebraclosed with respect to certain nonlinear commutation relations. This symmetry is better esized when the operators are realized in terms ofq-deformed derivatives.26 By expressing thetranslation and dilation operators asq-deformed derivatives, the scaling and difference operacan be mapped onto the generators of nonlinear algebra and the action of theq-deformed deriva-tives is extended to nondifferentiable functions. Moreover, such nonlinear algebras can beformed to the Fourier series generating algebra or can be mapped onto the slq(2) algebra.

After devoting Sec. II to definitions and notations, we present in Sec. III the deformationFourier algebra into a scaling function generating algebra, and identify a spectral problem wHaar dilation equation. In Sec. IV we introduce the definition of the most general scaling funand wavelet algebra. The uniqueness of the algebraic formulation of the dilation equatproved. A duality relation is obtained between any scaling function and certain nonlinear algwhich associates to any given scaling function~wavelet! an algebra and, for a given algebra ofcertain type, one finds the appropriate wavelet structure. In Sec. V we give examples of sfunction algebras for the Haar and theB-wavetes. We use some new limiting procedures in orto solve theq-difference and finite-difference equations. In Sec. VI we have used a varianextension of the deforming functional technique,28,29,31wherein we can obtain a mapping of thgenerators of nonlinear algebras to those of slq(2). Finally, Sec. VII comprises concluding comments, further extensions, and remarks.

II. BASIC ELEMENTS

An algebraic structure can be provided for any scaling function system and wavelet baorder to realize this construction we need three building blocks: the algebraic structureFourier system~starting point!, the wavelet theory~final aim!, andq-deformation~intermediatetool!.

A. Fourier algebras

In order to demonstrate our approach, we give an example for Fourier series, i.e., alhaving Fourier series as basis for their representation spaces.20 In the following we shall denote]kf /]xk5]kf 5 f k. The trigonometric~Fourier! system$uk.5eikx%kPZ diagonalizes all translationinvariant operators acting onL2(@0,2p#). We introduce three generators within a differentrealizationJ052 i ], J65e6 ix(2 i ])p, satisfying the commutation relations forp50, 1 only

@J0 , J6#56J6 , @J1 , J2#512~122i ]!p. ~1!

For p50 Eqs.~1! describe an algebra, denotedF 0 , that is isomorphic to an analytical prolongation, e(2,C), of the Euclidean algebrae(2,R) of rigid motions in the plane. The algebrae(2,R),generated byPx5Px

† , Py5Py† , and R52R†, with the commutators@R, Px,y#56Py,x ,

@Px , Py#50, is mapped through the generator mapping

J05 iR, J65Px6 iPy ,

onto theF 0 algebra. The unitary irreducible representations~unirreps! of F 0 are based on theeigenvectors of the self-adjoint operatorJ0 , J0un&5nun&. For any two distinct eigenstates,un& andun8&, by using the first commutator in Eq.~1!, we obtainn85n11 and hence the spectrum ofJ0

is unbounded, discrete and consists in equidistant eigenvalues. Hence the space of represeis generated by the Fourier system and the other generatorsJ6un&5un61& act like ladder opera-tors on theun& states, increasing/decreasing the scale by unity. Forp51 Eqs.~1! describe anotherLie algebra,F 1 , isomorphic with the symplectic algebra sp(2,R).su(1,1).so(2,1). This differ-


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2348 J. Math. Phys., Vol. 39, No. 4, April 1998 Ludu, Greiner, and Draayer

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ential realization has the same representation space asF 0 but it is not irreducible. The action othe generators ofF 1 is similar to that of the generators ofF 0 for nÞ0. Unlike theF 0 case,however, we haveJ6u71&5u0& and the subspaces$einx%nPN and $e2 inx%nPN are invariant sub-spaces ofF 1 . The algebrasF 0,1 suggest the possibility of other constructions in terms ofoperator] and the complex exponential functions of different scales.

B. Scaling functions and wavelets

Wavelets are able to reconstruct a signal through regular sampling, i.e., by analyzinsignal at different scales~which increase/decrease exponentially! with the step size between eacscale being the same. Different from the ordinary Fourier transform, which reproduces a fuas a superposition of complex exponentials, or from the windowed Fourier transform, wintroduces a scale into the analysis of signals, multiresolution analysis processes the signalusing the appropriate local scale. Basically, wavelets are constructed with a pair of operatodilation ~scaling! and finite difference~combinations of translations! operators, acting onL2(R)and defined byTa f (x)5 f (x1a), Db f (x)52b f (2bx), with a,b arbitrary real numbers. In thefollowing we will use the operatorDb522bDb for simplicity in equations~Db is the dilationoperator consistent with wavelet theory!. They are invertible, unitary, and fulfillTaDb

5DbT2ba, TaTb5Ta1b, DaDb5Da1b. The formal Taylor series of these operators areTa

5ea] and Db52beb ln 2x]. On the space of compact supported or rapidly decreasing functany holomorphic functionf (T) is locally polynomial. Indeed, iff is holomorphic, and its actionis taken on the compact supported function subspace ofFPL2(R), then the action off (T)5(kPZ CkT

k reduces to that of a Laurent polynomial by keeping only a finite number of termthe sum.

The wavelet system is given by a set of scaled and translated copies of a pair of functioscaling functionF and the mother waveletC. The basic fact about wavelets is that both thefundamental functions are finite linear combinations ofF, reflecting the self-similar character othe wavelet system. The defining equation for the scaling function~the dilation equation! is alinear finite-difference equation, including a scale change~q-difference27!

F~x!5D (k50

n

CkT2kF~x!5Dg~T!F~x!, ~2!

where the RHS sum is a polynomial inT,g(T). Equation ~2! is a fixed-point equation andconsequently it has only one unique solution.1–7,11,13The scaling functionF(x), as a solution ofEq. ~2!, is required to have two properties1–5

~1! *RF(x)dx51, the average value property;~2! ^Fn ,F&[*RF(x1n)F(x)dx5dn,0 , for anynPZ, the orthogonality condition.

These conditions introduce restrictions on the coefficientsCk in Eq. ~2!,1–4

(k50

n

Ck52, (k50

n

CkCk12l5d0l , l PZ. ~3!

The condition Eq.~3! yield a pattern ofL2(R) as a chain of subspacesVj,Vj 11 , each one beinggenerated by all the translations ofD jF, j PZ. By repeated application ofD and T on F,D jTnF(x)5F(2 j x1n)[F j ,n , one obtains a nonorthogonal basis inL2(R). The action of theoperator2T2lDg(2T21) on F, l being a unique odd integer, provides the mother wavefunction,

C~x!52T2lDg~2T21!F~x!52T2l(k

~21!kC2k11F~2x2k!. ~4!

The waveletC(x) has the property that$C j ,n% j ,nPZ is an orthonormal basis ofL2(R) Refs. 1–3andL2(R) is a direct sum of the orthogonal subspacesWj ~the orthogonal complements ofVj !,each of them generated by all possible translations ofD jC5C j ,0 with integral m:L2(R)


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5% j PZWj .1–5 Hence translated and dilated copies of the mother wavelet,C(2 j x1k), generate atrue orthonormal basis. Recursively applications of the Eqs.~2! and ~4! relate all the scalingfunctions and wavelets.

The simplest example is provided by the Haar wavelet, defined by the scaling funFHaar(x)51 if ux21/2u<1/2 and 0 otherwise. In this case we haveD(11T21)FHaar5FHaar andgHaar(T)511T21 ~C05C2151 and the rest 0 in Eq.~2!!. The corresponding Haar waveletCHaar5D(12T21)FHaar, i.e., l51 in Eq. ~4!.

There are similarities and differences between the Fourier and the wavelet approachesan algebraic point of view, in both approaches, there are eigenfunction equations, which kescale constant, and the ladder operators, which change the scale. The dilation equation,~2!,has its analog in the Fourier formalism, though it is a fixed-point equation and has onlysolution. The mother wavelet, Eq.~4!, has no analog in Fourier analysis. Each Fourier eigenfution carries one scale; a wavelet functionC(x) involves two scales, e.g.,F0,k andF2,k . Becauseof their localization, wavelets possess a degree of freedom beyond that of a Fourier snamely, the width of the support ofF. In the present approach this degree of freedom is assocwith the deformation parameter,q.

C. Quantum deformations

A possibility for constructing scaling functions/wavelets within an algebraic approachdeform the Fourier algebraic structure into a nonlinear system. In general, the scaling funand wavelets are not differentiable. This suggests the use of finite-difference operators insderivatives. Finite-difference operators are closed with respect to commutation but only wnonlinear algebraic constructions. Consequently, it is natural to useq-deformed derivatives to finda foundation for wavelet theory inq deformed algebras.

Quantum algebras refer to some specific deformations of Lie algebras, to which they rwhen the deformation parameterq is set equal to unity~for a recent monograph see Ref. 21, areferences therein!. The simplest example of aq-algebra is slq(2) whose Jordan–Schwingerealization is given in terms ofq-bosonic operators.26 Theq-deformed algebras have been appliin various branches of physics like spin-chain models, noncommutative spaces, rotationalof deformed nuclei, Hamiltonian quantization, and dynamical symmetry breaking. The basiment is theq-deformation of a certain objectx, which can be a number, an operator, or a functi

@x#s5qx2q2x

q2q21 ——→s→0

x, ~5!

whereq5es. The Taylor expansion ofT andD are related to theq-deformation of the derivativeoperator, according to the definition of the coordinate description of theq-deformed oscillatorintroduced in Ref. 26,

@]#sf 5Ts2T2s

2 sinh~s!f ~x!. ~6!

Analogously, we introduce the operator

@x]#s ln 2f ~x!5f ~2sx!2 f ~22sx!

2 sinh~s ln 2!5

Ds2D2s

2 sinh~s ln 2!f ~x!. ~7!

Whens→0, @]#s→] and @x]#s ln 2→x], Eqs.~6! and ~7! can be inverted and the occurrencethe q-deformed derivative and the translation/dilation operators becomes immediate

T6s51

2 S 6@]#s

h~s!1

@]#s/22

h2~s/2!12D , ~8!

D6s51

2 S 6@x]#s

h~s ln 2!1

@x]#s/22

h2~s ln 2/2!12D , ~9!


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where sPN ~or more generalPR! and h(s)51/(es2e2s). The q-deformed algebra slq(2),generated by$J0 ,J6%, is defined by theq-deformed version of the commutation relation

@J1 , J2#5@2J0#s ,

while the other two commutator relations remain undeformed, as for with sl(2). In addition to thistraditional version of slq(2), several generalized forms have been introduced, in two diffeways. First, by deforming only the commutator@J1 , J2# by using some arbitrary functionF(J0 ,q), @J1 , J2#5F(J0 ,q) independently proposed in Refs. 28, 29, and 30. The second wby deformations involving all three commutation relations, using two functionsG(J0 ,q) andF(J0 ,q)5@J1 , J2#, introduced in Ref. 31. Unlike the former, for which the spectrum ofJ0 islinear, the latter is characterized by an exponential spectrum forJ0 . Since in wavelet theory thedomains of analysis are divided exponentially, rather than linearly with bands of equal wsuch algebras are used in the following analyses.

III. HAAR SCALING FUNCTION ALGEBRA, As ,a

The aim in this section is to obtain operators depending onD and T, closed under somenonlinear algebra. This structure must provide an algebraic form for Eqs.~2!, ~4! and must bemapped into one Fourier algebra,F 0,1. Consider an operator depending onT and two realparameterss, a,

W0~s,a!5Tsa2T2sa cossp

2j~a!sinh s, ~10!

wherej(a)5@1/sinh(1)#sin(ap/2)22i cos(ap/2). The (s,a) parameters allowW0 to approach]or combinations ofT,

W0~0,a!5a

j~a!], ~11!

W0S 1

2,a D5

1

2 sinh~1/2!j~a!Ta/2, ~12!

W0~1,a!5Ta1T2a

2 sinh~1!j~a!, ~13!

W0~2,a!5T2a2T22a

2 sinh~2!j~a!. ~14!

In the limit s→0, W0 reduces to the normal derivative with respect tox, Eq. ~11!. In the limits5 1

2, we obtain a power of the translation operatorT, Eq. ~12!. Equation ~13! defines aq-deformation of unity, i.e.,W0(1,0)5@ i /4 sinh(1)#. W0(s,a) can be a Hermitian or antiHermitian operator,W0(1,a)56W0(1,a)†, depending ona. The last limit, Eq.~14!, represents afinite difference operator and is proportional to theq-derivative with respect to thea-deformation@2]#a , similar to Eqs.~6!–~9!. For a52k in Eq. ~12!, a5k in Eq. ~13! anda5k/2 in Eq. ~14!,the correspondingW0(s,a) are Laurent polynomials inT, providing a direct connection to thwavelet operators. For some values ofa, Eqs.~12!–~14! give invertible operators with respect tT,T5T(W0).

In order to introduce the dilation operatorD, we define

W6~s!5 12e

7 ix~s21!/2D7se7 ix~s21!/2~11T6s!. ~15!

In order to fulfil the Hermiticity condition (W1)†5W2 on L2(R) these generators can be redfined in the form

W6→W652217,s/2e7 ix~s21!/2D7se7 ix~s21!/2~11e2 i @s~s21!~112s!#/4T6s~2~s/2!~171!!.


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By using the commutators relations betweenD andT and by performing an integration by partone has a direct check of the relation (W1)†5W2 ,

^ f 1 ,W1 f 2&5ER

f 1* ~x!W1 f 2~x!dx5^W2 f 1 , f 2&.

From Eqs.~10! and ~15! one obtains the commutators,

@W0~s,a!, W1~s!#5G~s,T!W1~s!, ~16!

@W0~s,a!, W2~s!#52W2~s!G~s,T!, ~17!

@W1~s!, W2~s!#5F~s,T!, ~18!

with

G~s,T!5W0~s,a!2T2ssaeisa~112s!~s21!/22T22ssae2 isa~112s!~s21!/2 cossp

2 sinh~s!j~a!~19!

and

F~s,T!51

4~~11eis~112s!~s21!/2T2ss!~11T2s!2~11eis~1122s!~s21!/2~T222ss!~11Ts!!.

~20!

Equations~10!, ~15!–~20! describe a nonlinear associative algebra, generated byT6,D6, denotedAs,a . The eigenvalue problem forW6 provides the algebraic form for the scaling equation. WhW0(s,a) is invertible with respect toT, the algebra hasW0 ,W6 as generators which depend otwo parameters (s,a) and is homeomorphic with a specialq-deformed algebra, namely, thtwo-color quasitriangular Hopf algebraA6.31 The spectrum ofW0 is not equidistant becausF(s,T)Þconst3W0. The Casimir operator of the algebraAs,a is given by28–31

C5W2W11H~W0!5W1W21H~W0!2F~W0!, ~21!

where H is a real function, holomorphic in a neighborhood of 0 and which must satisfyfunctional equation

H~j!2H~j2G~j!!5F~j! ~22!

with j generic. The irreps ofAs,a are labeled by the eigenvaluesa of W0 and those of the Casimioperator. The commutator in Eq.~16! can be written in the form

W1W05~W02G!W1 . ~23!

If ua^Þua8& are eigenvectors ofW0 , with the corresponding eigenvaluesa,a8, we have from Eq.~23! a nonlinear recursion relation for all the eigenvalues

a85a2G~a!. ~24!

The algebraAs,a can be mapped intoe(2,C), the Lie algebra of a Fourier series, in the lims→0, a→2: W6→e6 ix5J6 , W0→2 i ]5J0 , where we take these limits in the orderAs0 ,a0

5 lima→a0lims→s0

As,a . The continuous mappingAs,a→A0,2.e(2,C) is an algebraic mor-phism and one can consider wavelets asq-deformed generalizations of the Fourier series, inabove sense.

As a first example, the Haar scaling function can be associated with a particular case ofAs,a ,i.e., the algebraA1,1. In the algebraA1,1 the eigenproblem forW2 provides the dilation equationfor the Haar scaling functionF(x) defined by Eq.~2! with g(T)511T21. We have in this caseG5W022W0

211, andW65 12D

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@W0 , W1#5~W022W0211!W1 ,

@W0 , W2#52W2~W022W0211!, ~25!

@W1 , W2#5 14~T2~W0!2T1/2~W0!2T21/2~W0!1T21~W0!!,

whereT(W0) is the solution of the equation 2W05T1T21, a pseudodifferential operator. ThCasimir operator ofA1,1 is a constant. Indeed, Eq.~22! for the general Casimir operator becom

H~W0!2H~2W0221!5F~W0!, ~26!

and has, in terms of theT5T(W0) operator, the form

x~T!2x~T2!51

4~T22T1/22T21/21T21!, ~27!

wherex(T)5H@(T1T21)/2#. The unique analytical solution for Eq.~27! reads

x~T!5 14~const2T2T1/22T21/2!, ~28!

which gives for the Casimir operator a constant.The spectrum ofW0 consists of periodic functions satisfying the equationF(x11)1F(x

21)52F(x). This takes one back to Fourier analysis. The way to the Haar scaling functionuse the spectrum ofW6 since inA1,1 the eigenproblem forW2 gives the dilation equation for theHaar scaling functionF(x). OnceF is obtained from 2W2F5F the generatorW0 carriesF intoan infinite sequence of functionsW0

nF. These functions are mutual orthogonal and generatespaceV0 . The action ofW6 on F yields the subsequenceW0

2nF. Consequently,V0 is an invariantspace for all the generators of the algebraA1,1.

The spectrum ofW0 depends on the values of the parameterss,a. In the case ofA1,1 thisrelation isa852a221. The corresponding representations are infinite-dimensional and the evalues ofW0u1,1 arean5cosh(2na), nPZ for any a. The corresponding eigenfunctions have tform uan&65e62nax which results in an exponential spectrum foran similar to the sequence oscales in wavelet theory. This is a self-similar spectrum with respect toa, like the sequence oscales in wavelet theory. A part of this spectrum is shown in Fig. 1. The action ofW6 is

W6uan&6511e62na

2uan61&6 . ~29!

Another basis for a representation ofA1,1 is given byuk&5e( ipx/k), kPZ. We have the action

W0uk&5cosp

kuk&, W6uk&5

1

2~11e6 ip/k!u261k&,

and on this basis (W1)†5W2 . There are two invariant spaces,u2k& and u22k&, kPN, withW6u0&51.

The same procedure can be followed for any set of the parameterss,a. For example, if wechoose the algebraA2,1/2 with W0 defined in Eq.~12!, we obtain the spectrum ofW0 described bythe recursion relation

a85aS a1Aa21111

a1Aa211D , ~30!

which also gives a nonlinear, unbounded, discrete representation ofA2,1/2.The last thing to prove is the uniqueness of the two-scale equation in theA1,1 scaling function

generating algebra. The two-scale equation, Eq.~2!, in its algebraic form 2W2F5F is not unique


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in A1,1 if there exists an operator, similar with that occurring in the dilation equation, wcommutes withW2 . This operator should contain higher powers inT and consequently is anelement ofU(A1,1).

Proposition 1: In U(A1,1) there exists a unique operator X5Dx(T21) such that@W2 , X#50. The function x(j) is an integer, it is not a polynomial and it is unbounded, for genericj.

Proof: We take forX a Laurent seriesx(T21)5SkPZ CkT2k then the condition@W2 ,X#

50 results in a recursion relation for the coefficientsCk ,

C2k115Ck2C2k21 , C2k5Ck2C2k22 , ~31!

for any kPZ. Equations ~31! have one trivial solution which reproducesW2 ,x(T21)5C21(11T21) and only one different solution, withCk56C21 or 0, uniquely defined~C25C635C245C55C265C75C285C211•••50, C225C255C65C295C2105C105C115•••5C21 and C15C45C275C85C95C2125C125•••2C21 , etc.!. The sequence of nonzercoefficients is infinite, and$Ck% is not a Cauchy sequence.~q.e.d.! We note that theA1,1 algebrais also a Hopf algebra,21,30,31defined by Eqs.~25! and by the coproduct, counit and antipode in tform,

DT65T6^ T6, DD65D6

^ D6, e~T6!5e~D6!51,

S~T6!5T7, S~D6!5D7.

All its generators are primitive elements.

IV. GENERAL SCALING AND WAVELET ALGEBRA

This algebraic approach for the Haar scaling function can be generalized to yield a nonalgebra for any scaling function, and conversely, to find the dilation equation and scaling funfor a certain type of algebra. The procedure starts with an algebra generated by dilatiotranslation and constructs, within this algebra, the two-scale equation. The generators of thAs,a

algebra can be generalized as

W0→ j 0~T!, W6→ j 6~D,T!5e7 ix~s21!/2D7se7 ix~s21!/2j ~T61!, ~32!

FIG. 1. The dependence ona0 of three eigenvaluesan of W0 in A1,1, for n52 ~the largest period!, 4 and 6~the smallestperiod!. The three spectra have self-similar structure fora0P@0, 1#.


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with j 0(T) and j (T) being arbitrary functions ofT ands, holomorphic in a neighborhood ofT51, with their dependence ons being such that in the limits→0, j 0(T)→2 i ], j (T)→1. In thiscase the commutation relations, Eqs.~16!–~20!, become

@ j 0 , j 1#5Gj 1 , @ j 0 , j 2#52 j 2G, @ j 1 , j 2#5F, ~33!

with G,F depending onT through j 0 , j , respectively,

G5G~s,T!5 j 0~s,T!2 j 0~s,T2sei ~112s!~s21!/2!, ~34!

F5F~s,T!5 j ~s,ei ~s21!/2~112s!T2s! j ~s,T21!2 j ~s,T222s

ei ~s21!/2~1122s!! j ~s,T!. ~35!

Equations~33!–~35! define a nonlinear algebra denotedA j 0 , j as a generalization ofAs,a . If thefunction j (T) is invertible with respect toT, A j 0 , j can be expressed in terms of the generat

j 0 , j 6 only and thenG(T)5G ( j 0), F(T)5F ( j 0). The algebraic morphismj 0(T)→W0 andj (T)→11Ts provides the mappingA j 0 , j→As,a . Since the first two commutators ofA j 0 , j , Eq.~33!, do not depend on the functionj (T), and the third commutator in Eq.~33! does not dependon the functionj 0(T), the closure condition forA j 0 , j is independent of the functionsj 0(T) andj (T) and hence they can be chosen in a convenient way to provide any dilation equationform of the eigenproblem forj 2 . The algebraic closure conditions forA j 0 , j , together with thedefinitions of j , j 0 require

j 0~T2!2 j 0~T!5G ~ j 0~T2!!, ~36!

j ~T2! j ~T21!2 j ~T21/2! j ~T!5F ~ j 0~T!!. ~37!

Both Eqs.~36! and ~37! are nonlinear, and in general, difficult to solve analytically. The uniqsolution for the dilation equation arej 6 since the Casimir operator of this algebra depends oTonly. Choosingg(T)5 j (T) yields the dilation equation in the form 2j 2F5F. In the limit ofAs,a this provides again the Haar two-scale equation. The corresponding scaling function bto the basis of the representation ofj 2 with eigenvalue 1/2. The remaining arbitrary functionj 0

can now be selected to obtain the wavelet generating operatorj 2 j 0F5C, in agreement with Eq.~4!. One can formally construct a basis of the representation forj 2 with the functionsun&5(ln T)nF. The procedure is the following: for a given algebraA j 0 j ~therefore given functionsG ,F ! solve Eqs.~36! and ~37! with respect to the functionsj 0 , j 6 and obtain the correspondindilation equation in the form 2j 2F5F. A certain combination of generators produces the cresponding wavelet equation forC. Conversely, given a scaling/wavelet function, and conquently its dilation equation andg(T), one can choosej 25Dg(T) and then solves the equatioDg(2T21)5 j 2 j 0 with respect toj 0 . With j 0 , j 6 known one can solve Eqs.~36! and ~37! withrespect toG , F to construct the algebra.

This algebraic approach is in some sense universal, what is modified is the specific realof the nonlinear algebra in terms ofD and T. Constraints are imposed by the two limitinapproaches~two-scale equation and Fourier limit!. Loosely speaking, the closure of the algebprovides the fixed-point dilation equation and the nonlinearity of the algebra provides thenential scaling.

We also note thatV05$(kCkTkF%, i.e., the space generated by all the integer translation

F, and any otherVj5D jV0 , is not invariant to the action ofj 2 . For example, the conditionj 2DF5(kCkT

kDF implies the existence of an operator containingD, whose commutator withj 2 is a function ofT only. From the definition of the algebra this is impossible, and consequethis proves the above conjecture. TheVj spaces form, by recursion, a basis inL2(R).

In order to prove the uniqueness of the two-scale equation in the general case, we agProposition 1, forW2→ j 25D j (T21)5D(kPZ j kT

k. We have to solve the commutation eqution @D j (T21), X)] 50 for x(T)5(kPZXkT

k, X5Dx(T). This implies that the arbitrary functions x(j) and j (j) must fulfill the functional condition~j a generic variable!

x~j2! j ~j!5x~j! j ~j2!. ~38!


nd 1.

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func-creasingvided

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Equation~38! does not carry any restriction with respect to the values of the functions in 0 aSinceD j (T) represents a scaling operator we have, according to the dilation equation,j (1)51,j (21)50 which impliesx(21)50. If F is the corresponding scaling function forj 2 thenXF isalso an eigenfunction ofj 2 , j 2(XF)5XF. In order for Dx(T) to satisfy the average valuproperty~1! in Sec. II B, we impose the additional conditionx(1)51. The last condition requiresDX(T), the orthogonality condition~2! of Sec. II B, be equivalent to the condition1–4

ux~T!u21ux~2T!u251. ~39!

By introducing Eq.~38! into Eq. ~39! we obtain X5 j 2 which provides the trivial indentitysolution, and hence the uniqueness.

V. CONSTRUCTION OF THE SCALING AND WAVELET ALGEBRA

The link between the scaling function and wavelet, and the corresponding scaling algesupported by the solutions of Eqs.~36! and~37!. In the following we present a method for solvinthese equations, from the wavelets towards the algebra. The dilation equation Eq.~2! has the formof a fixed-point equation. Therefore we must try to find its eigenvectors as limits of sometional sequences. The limits of these sequences should be compact supported or rapidly defunctions for two reasons; to obtain scaling functions with good localization, and to procorrect behavior of the action of the operatorsf (T). Since the scaling function will be expresseas a limit of a sequence, one has to look for operators which commute with the limit~are absorbedin the limit!. We choose a test functionD0(x) and a sequence of operatorsf n from the universalcoveringU(A j 0 , j ) of the scaling function algebra introduced in the preceding section, suchthe limit F(x)5 limn→` f nD0(x) exists in the weak topology and it provides the scaling functi

In the following we show how to construct the scaling algebraA j 0 , j from the dilationequation, written in the formj 2(T,D,s)F5F. We have the following proposition in the framework of the algebraA j 0 , j , for s51.

Proposition 2: Let fn( j 0) be a functional sequence in U(A j 0 , j ), s51, D0(x) a (test) function

and F(x) the scaling function. If the limitF(x)5 limn→` f n( j 0) j 2n D0(x) exists, and

limn→`@ f n( j 02G (s, j 0))/ f n21( j 0)#51, then j2F(x)5F(x).Proof: From the RHS of the first condition in the hypothesis and from Eq.~33! we have

f n~ j 0! j 2n 5 j 2 f n~ j 02G ~s, j 0!! j 2

n215 j 2

f n~ j 02G ~s, j 0!!

f n21~ j s!f n21~ j 0! j 2

n21.

It follows

j 2F5 limn→`

j 2 f n~ j 0!D0~x!5F,

q.e.d. It follows from Proposition 2 that for a given dilation equationj 2F5F, ~both j 2 andFgiven! we can find out a sequencef n( j 0), the operatorG (s, j 0) and a test functionD0(x), suchthat these objects satisfy the hypothesis of Proposition 2. Then it follows that one can constrscaling algebra, since withj 0 andG (s, j 0) found,F (s, j 0) results from the last commutator in Eq~33!. In general, one starts withj 0 as an arbitrary function ofT which can be deformed into2 i ]when mappingA j 0 , j to F 0,1. As j 2 is provided by the two-scale equation,j 2F5F, othersolutions are forbidden. If limn→` f n( j 0)5 f `( j 0)Þconst., this limit should commute withj 2

which is forbidden by Proposition 1. This procedure closes the construction of the algebraA j 0 , j .In the following we give an algorithm for findingf n( j 0) and j 0(T) and illustrate it with two

examples. Since any mother scaling function is defined by the dilation equation~Dh(T)F5F orj 2F5F! it is natural to search for solutions by using a recursion algorithm. For instance, wfind a test functionD0(x) and a triple of operatorsA,B,C such that the limit F(x)5B limn→` AnD0(x) exists and in additionCB5BAk, for a finite positive integerk. Then, wehave the property CF(x)5F(x). Indeed, CF5CB limn→` AnD05B limn→` An1kD0

5B limn→` AnD05F. From the dilation equation we know thatC should have the formC


let.

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nction,w in


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5Dc(T) with c(T) a function ofT. One of the simplest choices is to useA5D and B5b(T).Then we haveCB5Dc(T)b(T)5c(T1/2)b(T1/2)D5b(T)A; that is,k51 and a restriction for thearbitrary functionsc(T) andb(T) arises,

c~T1/2!b~T1/2!5b~T!. ~40!

This equation is useful in both directions~algebraic↔dilation equation!, since one can start witha given scaling function~given c(T)! and find the operatorb(T) involved in the algebra andconversely.

Now consider the algorithm for the Haar scaling function,c(T)511T21. We look forsolutions of Eq.~40! as Laurent series forb(T)5(kPZbkT

k. In this case Eq.~40! has an uniquesolution, b(T)5const.•(12T21). This gives again a unique solution forF5(12T21)limn→` DnD0 . We stress thatDb(T) is exactly the operator which gives the Haar waveFurther, we can expressFHaar in the formF(x)5H(x)2H(x21), whereH(x) is the Heavisidedistribution. For any sequence ofC` functions dn(x)→d(x), we have Dn(x)5*dn(x)dx→H(x) and we can write

F~x!5 limn→`

~Dn~x!2Dn~x21!!. ~41!

For example, we can use the sequencesdn5(1/p)@22n/(x21222n)# or dn5@2n21/cosh2(2nx)#and Dn5@arctan(2nx)/p# or Dn5 1

2 tanh(2nx), respectively. The latter example is a soliton-likshape, having good localization. These sequences converge tod(x), respectively. In this way wehave selected subsequences that step in powers of 2. Equation~41! can be written as

F~x!5 limn→`

~12T21!DnD0~x!. ~42!

We can express this definition in terms of the algebraA1,1. By using Eq.~25! and the propertiesof the operatorsD andT, we can write Eq.~42! in the form

F~x!5 limn→`

~12T222n!~2W2!nD0~x!. ~43!

Indeed, from the commutation relation betweenD andT and the definition ofW2 we have

~2W2!n5~D~11T21!!n5Dn~11T211•••1T22n11!

5~11T222n1~T222n

!21•••1~T222n!2n21!Dn. ~44!

Hence, we can write (12T21)Dn5(12T222n)(2W2)n Moreover, we can write, by using th

inverted form forW2(T), the dilation equation in a pure algebraic form,

F~x!5 limn→`

~12T222n~W0!!~2W2!nD0~x!. ~45!

From this last equation it follows that 2W2F5F, i.e., the Haar dilation equation. If the functioD0(x) is chosen from a class of suitable functions for a wavelet analyzis then its scaling funF(x), Eq. ~45!, gives a rapidly convergent wavelet expansion. In order to exemplify we shoFig. 2 scaling functions obtained by this procedure for different values of the parameters withinAs,a . For s50 and 1 this functions yields the Haar scaling function.

We give another example for theF2 B-scaling function and theC2 B-wavelet. We define

F2~x!5H 0 x<0, x>1

4x 0<x<1/2

24x14 1/2<x<1

, ~46!

or


et

spec-sed toobtain


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F2~x!5 limn→`

1

22n11 ~122T21/21T21!DnD0,tri~x!, ~47!

with

D0,tri~x!52x

parctan~2x!2

ln~114x2!

2p5E

0

2p

D0~x!dx. ~48!

Following the above algorithm we obtain the corresponding dilation equation,

j 2u triF2~x!5D

2~11T21/2!2F2~x!. ~49!

By using the commutation relation]D52D] and the dilation equation Eq.~49! we can obtain aself-contained equation for the wavelet,

C tri~x!5D~112T21/21T2!2C tri . ~50!

The corresponding algebras for Haar andF2 B-scaling functions are different. For the B-wavelwe have, inA1,1,

G~W0!5W02W022 1

2. ~51!

This new algebra, fulfills different commutation relations and consequently has a differenttrum for W0 , not the Haar scaling algebra. The algorithm introduced above can also be ugenerate other scaling functions and their corresponding algebras. An interesting line is to

FIG. 2. The continuous deformation of the scaling functionF in As,a as function of the parameters. For s50,1 this isthe Haar scaling function.


e B-

ions

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of

. In the


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polygon-like wavelets or, by following the same procedure which guided us to obtain thscaling function from the Haar scaling function, to obtain smoother scaling functions.

VI. THE slq„2… LIMIT OF Aj 0 ,j

The quantum group slq(2) or its extensions21–26,28–31are associative algebras overC gener-ated by three operators,J05(J0)†, J1 , andJ25(J1)†, satisfying the commutation relations,

@J0 , J1#5J1 , @J0 , J2#52J2 , @J1 , J2#5@J0#s , ~52!

where@J0# is a deformation ofJ0 , that is, a real, parameter-dependent function ofJ0 , holomor-phic in the neighborhood of zero, approaching 2J0 in the limit s→0. We have to find a deformingfunctional that transforms theA j 0 , j generators into operators satisfying the commutation relatEq. ~52!. For this purpose the first equation in Eqs.~33! can be written as

~ j 02G ~s,J0!! j 15 j 1 j 0 , ~53!

and hence, for every entire functionp(j), we can write

p~ j 02G ~s, j 0!! j 15 j 1p~ j 0!. ~54!

Let us consider the functional equation:

G~j2G ~s,j!!5G~j!21 ~55!

for a given functionG~j!, with j generic. If this equation has a solutionG~j! that is an entirefunction, then Eq.~55! can be written, forp(j)5G(j), in the form@G( j 0) j 1#5 j 1 , and corre-spondingly@G( j 0), j 2#52 j 2 . These equations give the correspondence between the waalgebras and theq-deformed algebras like slq(2) or other extensions of them. They map tstructure ofA j 0 , j 6

, Eq. ~33!, onto the structure described by the first two equations in Eq.~52!.For the third commutator of each of these algebras we use the functionG~j!, which allows Eqs.~33! to be reduced to the third equation in Eqs.~52! through the mapping,

J05G~ j 0!, J65 j 6 . ~56!

If G is invertible, the third equation in Eqs.~33! in A j 0 , j can be reduced to the third equationEqs. ~52! of an algebra defined by the function@J0# with the identificationF +G215@ # where@ #+G means the composition of the two functions, i.e., (@ #+G)(j)5@G(j)5@G(j)#. In the case ofslq(2), i.e., @J0@5@2J0#q , the functionF (s, j 0 ,s) becomes

F ~s, j 0!52~f~ j 0!!22~f~ j 0!!22

q2q21 , ~57!

where q5es,f(j)[q2G(j) has to satisfy the equationf(j2G (s,j))5qf(j). Consequently,Eqs. ~53!–~55! provide the reduction of slq(2) into A j 0 , j , throughG. For details of the abovetechnique of mapping and reduction of nonlinear algebras one can see examples in Ref. 31case of the algebraA1,1 Eqs.~53!–~55! give the condition

p~W021!5W02G~1,W0!, ~58!

which has the solution

J05p~W0!uA1,15cosh~22W0b!, ~59!

for any arbitrary constantb. By inverting Eq.~59! we obtain

W0571

ln~2!lnS ]

bD . ~60!


ctns-,n

f

ferent

ie

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ropriatendings. The


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Conversely, starting from slq(2) and its deformation@ #q and mapping it intoA1,1 we obtain forthe operatorF(s,T)5F(s,T(W0)),

F5 f ~G~W0!!5 f ~p21~W0!!5@2p21~W0!#. ~61!

A diagram containing these algebra morphisms is presented in Fig. 3.

VII. FURTHER EXTENSIONS, COMMENTS, AND CONCLUSIONS

In this paper we present a method for connecting classes of nonlinear~q-deformed! algebras,with scaling functions and wavelets. This connection is reciprocal. The algebraA j 0 , j us51 allowsthe choice of different dilation equations, i.e., for different functionsj (1,T) one can obtain dif-ferent scaling functions. The functionj 0(1,T) should be chosen in order to give the correwavelet function through the action ofj 0 j 2 . The connection with quantum groups is more traparent if we choose forj 05@]#s51/25W0(2,1/2), Eq.~14!, and forF, the Haar scaling functionj 25D(11T21). The action of theq-derivative@]#1/2 on F gives exactly the Haar wavelet. Ithis example the operatorj 0 is anti-Hermitian and we can identify its spectrum~imaginary eigen-values! from the commutation relations. We also have the relationC5( j 02G(1,j 0)F, so thatone can express the eigenvectorsua& of J0 in the wavelet basis of the algebra. Ifua&5( j ,n Cj ,nF j ,n , we obtain a recursion relation for the coefficients ofua& and for the eigenfunc-tions of j 0 (k Cj ,p22 j k

Ck5aCj ,p for any j andp integer, whereCk are the Taylor coefficients othe function2j2lg(2j). We note that by using theq-derivative instead of a function ofT weobtain exactly the same mother wavelet~Haar wavelet!.

We want to note a natural extension of the above developments, coming from a difrealization of sp(2,R) in terms of the full algebra of symmetry of the real line:X05x] ~dilations!,X25] ~translations!, andX15x2] ~expansions!. This is the maximal finitely generated real Lalgebra with generators in the formxn]. By exponentiation, the generatorsX0 ,X2 provideD andT and the last generator has the actionEa5eaX1 f (x)5 f (x/12ax). For a.0 the action corre-sponds to a contraction of the function and for21,a,0 it is a dilation. But, fora,21 anduau>1 we obtain a splitting of the function in two Heaviside distributions,Eaf (x)5H(x)1H(2x11/a). A possible closed nonlinear algebra containingD, T, andE could be an oppor-tunity for wavelets with multiscale properties.18,19

In conclusion, we found that the operators of dilation and translation (D,T) can be combinedin such a way as to generate nonlinear algebras, depending on certain parameters (s,a). We haveinvestigated these algebras from the point of view of quantum groups, discussing their unCasimir operators, and reductions of these algebras to otherq-deformed algebras, like slq(2) or tothe Fourier series generating algebra. It has been shown that such algebras provide an appframework for the foundation of the wavelets analysis and for the obtaining the corresposcaling functions. We have worked out two examples, the Haar and the B-scaling function

FIG. 3. A diagrammatic representation of the algebraic maps~arrows! and morphisms~.! of A j 0 j , As,a , slq(2), sl~2!,andF 0,1.


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algorithm provides a general algebraic method for finding specific scaling function. Direct acations of such an approach can found in the theory of finite-difference equations andq-differenceequations.21–25,27

This study represents only a first step towards understanding the relation between theof nonlinear algebras~having exponential spectra! and wavelets with their finite-difference equtions. We conjecture that such an algebraic approach~nonlinear algebras! to scale invariant struc-tures can lead to interesting mathematical constructions like tools for identifying isolated cohstructures associated with certain mother wavelets and limiting nonlinear algebraic structuretranzition between such self-organized structures can be carried out through the modificathe deformation parameterq, with the intermediate domain reprezenting ‘‘noise’’~nonclosedalgebraic structures!.

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nd


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