CONSTRUCTION OF ORTHONORMAL WAVELETS USING KAMPÉ DE FÉRIET FUNCTIONS

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 1 Number 1 Pages S X Article electronically published on May 1 CONSTRUCTION OF ORTHONORMAL WAVELETS USING KAMPÉ DE FÉRIET FUNCTIONS AHMED I. ZAYED Communicated by David R. Larson Abstract. One of the main aims of this paper is to bridge the gap between two branches of mathematics special functions wavelets. This is done by showing how special functions can be used to construct orthonormal wavelet bases in a multiresolution analysis setting. The construction uses hypergeometric functions of one two variables a generalization of the latter known as Kampé defériet functions. The mother wavelets constructed by this process are entire functions given by rapidly converging power series that allow easy fast numerical evaluation. Explicit representation of wavelets facilitates among other things the study of the analytic properties of wavelets. 1. Introduction Special functions of mathematical physics such as Legendre Hermite Laguerre hypergeometric functions have been around for more than two centuries have been used successfully in many applications in applied mathematics physics engineering. On the other h wavelet analysis has emerged in the last decade not only as another powerful mathematical tool for applications but also as a deep profound mathematical theory. Up until now these two branches of mathematics special functions wavelets have not had much in common. One of the main aims of this paper is to bridge this gap by showing how special functions can be used to construct orthonormal wavelet bases in a multiresolution analysis setting. The construction uses hypergeometric functions of one two variables a generalization of the latter known as Kampé defériet functions. The mother wavelets constructed by this process are entire functions given by rapidly converging power series that allow easy fast numerical evaluation. Moreover explicit representation of wavelets facilitates among other things the study of the analytic properties of wavelets. The paper is organized as follows. In Section we introduce the notation special functions that will be used in the sequel. For lack of space we shall assume that the reader is familiar with wavelets multiresolution analyses. The main result is presented in Section as a theorem. Received by the editors November 8. Mathematics Subject Classification. Primary 4C4 C; Secondary 4C15 E. Key words phrases. Orthonormal wavelets blimited wavelets multiresolution analysis special functions hypergeometric functions Kampé de Fériet functions. 89 c American Mathematical Society

2 894 AHMED I. ZAYED. Preliminaries We adopt the following Pochhammer notation: a 1 a n aa +1...a + n 1 Γa + n Γa where n 1... a is a complex number. The generalized hypergeometric series p F q is defined as [ ] a1... a p ; z pf q b 1... b p F q a 1... a p ; b 1... b q ; z q a 1 n a n...a p n z n z is complex. b 1 n b n...b q n n! n The constants a i i 1... p b j j 1... q are complex numbers none of the b j s is a negative integer. The series converges for all finite z if p q converges for z < 1ifp q +1 converges for z 1ifp q +1 Re { q i1 b i p i1 a i} > diverges for all z ifp>q+1. Hypergeometric series in two variables can be defined similarly. We shall introduce the following generalization of the hypergeometric function of two variables which was originally introduced by Kampé defériet see [ Ch. 1] slightly generalized by Srivastava Pa see [6 p. 6]. The generalized Kampé defériet function of two variables is defined as F p:q;k l:m;n a p:b q ; c k ; α l :β m ; γ n ; x y rs p j1 a q j r+s j1 b k j r j1 c j s x r y s l j1 α m j r+s j1 β n j r j1 γ j s r! s! where N j1 z j z 1 z...z N none of the parameters in the denominator is a negative integer. The series converges for all x y if p+q <l+m+1 p+k < l + n +1. If p + q l + m +1p + k<l+ n + 1 then the series converges for x 1ally provided that l m p q Re α j + β j a j b j >. j1 j1 If p + q<l+ m +1 p + k l + n +1 then the series converges for y 1 all x provided that the same condition holds but with the β s replaced by γ s the b s replaced by c s. Young s function [7] of order ν ν which will be denoted by Y ν is defined by 1 Y ν z z ν k 1 k z k Γν +k +1 j1 j1 z ν Γν +1 1 F 1; ν +1 ν + ; z. 4 This function should be distinguished from the Bessel function of the second kind order ν which is usually denoted by Y ν. Clearly Y z cosz Y 1 z sin z.

3 CONSTRUCTION OF ORTHONORMAL WAVELETS 895 Another important related special function that will be used in the sequel is the following integral of Young s function I να defined by I να x x α 1 k x k+ν+α+1 Y ν x dx k + ν + α +1Γν +k +1 k x ν+α+1 ν + α +1Γν +1 F 1 ν + α +1 ; ν +1 ν + ν + α + ; x. 4 In particular x cos xdx I1/x x sin xdx I11/x. Here we assume that the reader is familiar with the theory of wavelets multiresolution analysis; see [1] for details. It is well known that a function φt is an orthogonal scaling function of a multiresolution analysis if it satisfies the following conditions: i k φω +k 1 ii φω 1 k c ke ikω/ φ ω m ω φ ω 4 where m ω 1 k c ke ikω/. iii φω is continuous at ω φ 1. The mother wavelet ψ can be obtained from the relation ω ω 5 ψω e iω/ m + φ.. The main result In this section we present the main result. But first we need the following lemmas. The first one whose proof can be found in [8 9] will play an important role in constructing the mother wavelets. Lemma 1. Let h be a function satisfying the following conditions: 1 h L 1 R h hxdx 1 4 hx is even 5 support h [ ]. Let w+ 6 gw hxdx. w Then gw is a nonnegative even continuous function with support in [ 4 ] 4 gw 1on [ ]. Moreover k gw +k 1. The function ˆφw gw is an orthogonal scaling function of a multiresolution analysis.

4 896 AHMED I. ZAYED The scaling function constructed in the above lemma is clearly blimited; hence so is its associated mother wavelet. This method of constructing blimited wavelets is related yet different from the one developed by Hernez Weiss [5 Ch. ]. Lemma. Let I Iu z J Ju z u u 1 x coszx dx 1 x sinzx dx where u 1/ z is real. Then I u 1 u F :; : 1/ 1; ; 1:; A B / : ; ; zu 1 J u F :;1 : 1/ 1; 1; 1:;1 A B : ; /; where A u /u 1 B u z /4. In particular if u 1/ 7 I 1 F :; : 1/ 1; ; 1:; / : ; ; 1 z /8 8 J z : 1/ 1; 1; 4 F :;1 1:;1 : ; /; Proof. Exping coszx in a power series leads to 1 n z n u I 1 x x n dx. n! n 1 z /8. Interchanging the summation integration signs is permissible since the series converges uniformly for all x z real. Setting x v weobtain I 1 n 1 n z n n! u of formula in [4 p. 84] which states that 9 u 1 vv n 1/ dv x µ 1 uµ dx 1 + βx ν µ F 1 ν µ;1+µ; βu Reµ > arg 1 + βu <

5 CONSTRUCTION OF ORTHONORMAL WAVELETS 897 we have I n Using the transformation [ p. 64] we get I 1 u 1/ 1 n z n u n+1 F 1 1 n +1! n+ 1 ; n + ; u. F 1 a b; c; w 1 w a F 1 a c b; c; w/w 1 n 1 n z n u n+1 F 1 1/ 1; n +/; u /u 1. n +1! Exping the hypergeometric function in a power series we have 1 I 1 u 1/ n 1 n z n u n+1 n +1! 1/ k 1 k n + k k! Ak where A u /u 1. The expansion is valid for A < 1. In fact since the hypergeometric series of F 1 a b; c; w converges absolutely uniformly for w 1 provided that Rec a b > it follows that for u 1/ we can interchange the summation in 1 to obtain I 1 u 1/ 1 n z n u n+1 1 k 1 k n +1! n + k k! A k. nk By using the Legendre duplication formula n + 1! Γn + n+1 Γn +1Γ n + n n!/ n noting that k we have n +/ k Γn + k +/ Γn +/ I u1 u 1/ / n+k Γ/ Γn +/ nk 1 k 1 k B n A k / n+k n! k! / n+k / n where B z u /. Thus using the notation of Section we have I u1 u 1/ F :; 1:; / : ; ; A B. The right-h side is well defined even for A 1; see Section. As for J we exp sinzx inapowerseriestoobtain J n 1 n z n+1 n +1! u 1 x x n+1 dx 1 n 1 n z n+1 n +1! u 1 vv n dv

6 898 AHMED I. ZAYED which with the aid of 9 yields 1 n z n+1 u n+ n +1!n +1 F 1 1 n+1;n +;u J 1 n 1 n z n+1 u n+ n +! n F 1 1 n+1;n +;u. Exping the hypergeometric function in a power series we have But J 1 u 1/ 1 u 1/ n kn 1 n z n+1 u n+ F 1 1/ 1; n +; n +! 1 n z n+1 u n+ n +! 1/ k 1 k A k. n + k k! n +! Γn + 1 n+ Γn +Γn +/ u u 1 n+1 Γn +/ n n + k Γn + k + Γn + n+k ΓΓn + ; hence J 1 u 1/ kn zu 1 u 1/ where B u z /4. Therefore zu J 1 u 1/ F :;1 1:;1 1 n z n+1 u n+ 1/ k 1 k A k n+1 n+k / n k! kn 1/ k 1 k 1 n A k B n n+k / n k! n! : 1/ 1; 1; A B : ; /;. Now we state prove our main theorem. Theorem 1. The functions φ 1 t φ t given by 11 φ 1 t sin t t [ ] t t t t + Y t / / cos Y 5/ sin

7 CONSTRUCTION OF ORTHONORMAL WAVELETS 899 φ t { } cos4t/ sin t cos t t + t t + 1 t cos F :; : 1/ 1; 1:; 1 t /6 / : ; ; + sint/ t 1 t 18 F :;1 : 1/ 1; 1; 1:;1 1 t /6 : ; /; are orthogonal scaling functions of a multiresolution analysis their associated mother wavelets are given respectively by ψ 1 t + 1 { / t t t t cos I t / 1/ sin I 11/ 1 + cos 8 ψ t + 1 { sin t + t + 1 4t cos 1 + t 18 sin where τ t Φτ F :; 1:; 4t Y / 4t cos t cost/ t + [Φτ+Φτ] 4t [Ψτ 4Ψτ] : 1/ 1; ; / : ; ; : 1/ 1; 1; Ψτ F :;1 1:;1 : ; /; sin 4t Y 5/ 4t cos8t/ cos t t 1 τ 8 1 τ 8. + } } sin t t Proof. We start with φ 1. In Lemma 1 let ht χ [ ]t be the characteristic function of the interval [ ] define gw w+ w htdt. With straightforward calculations one can show that if w 4 gw w + if 4 w 1 if w w + if w 4.

8 9 AHMED I. ZAYED Since gw is nonnegative by Lemma 1 we define ˆφ 1 w gw; hence 14 ˆφ 1 w +k gw +k k k w+k+ htdt k w+k htdt 1. Thus {φ 1 t n} is orthonormal in L R. If we define m w asthe4-periodic extension of ˆφ 1 w it will follow that ˆφ 1 satisfies the dilation equation 4ii as well as 4iii. Therefore φ 1 t is an orthogonal scaling function of a multiresolution analysis. To obtain φ 1 in closed form we use the inversion formula for the Fourier transform which in view of the fact that ˆφ 1 is even yields 4 φ 1 t 1 ˆφ 1 we itw dw 1 ˆφ 1 wcoswt dw { 1 4 } coswt dw + w coswt dw sin t + tγ γ cos dγ. t By setting w γ 1c t weobtain 15 φ 1 t sin t + t sin t + t But from [9] 1 [ cos c 1 1 w cos[cw +1]dw 1 1 w coscw dw sin c 1 1 w sincw dw ] w coscw dw 1 w sincw dw Y / c c / Y 5/ c c / the substitution of 16 into 15 yields 11. To derive ψ 1 t explicitly we appeal to Eq. 5. Because of the symmetry of e iw/ ˆψ1 w it suffices to consider its restriction to the positive real axis that is given by e iw/ ˆψ1 w w / w 1 / w 4/ w 4/ w 8/ 4 + 8/ w.

9 CONSTRUCTION OF ORTHONORMAL WAVELETS 91 By taking the inverse Fourier transform of e iw/ ˆψ 1 w we obtain ψ 1 t / e iw/ ˆψ1 wcostw dw { 1 4/ w 8/ } 1costw dw + w +costw dw / 4/ 4 γt γ 1cos dγ + 4 4γt γ cos dγ u cosαu + α du + 1 u cosβu + β du where α t/ β α. Thus in view of Eq. we have ψ 1 t + 1 { 1 1 cos α u cosαu du sin α u sinαu du 1 1 } + cos β 1 u cosβu du sinβ 1 u sinβu du { 1 cos αi1/ α sin αi α / 11/ α + cos βy/ β sin βy β / 5/ β } which is 1. As for φ wetake ht { 9 x + if x 9 x + if x. Then it is easy to verify that h satisfies all the conditions of Lemma 1. Set gw w+ w htdt. With some easy calculations we have w 4 9 w + 1w +8 4 w 9 w 6 w 1 w gw 1 w 9 w + 6w 1 w 9 w 1w +8 w 4 4 w. It follows as before that φ t is an orthogonal scaling function of a multiresolution analysis. To obtain φ t explicitly we have φ t 1 ˆφ we itw dw 1 4 ˆφ wcostw dw 1 costw dw + I + I sint/ t + I + I

10 9 AHMED I. ZAYED where I 1 9 w + 6w 1costw dw I 1 To evaluate I wehave 4 I 4 w 8 w [ cos4t/ t w 4 + sin t t 9 w 1w +8 costw dw. costw dw costw dw ] cos t t. Evaluating I is more difficult. To this end we have I 9 1 w costw dw 9 γ cos t 1 x cos τx + t γ + dγ 4 w 4 costw dw [ dx cos t ] t I sin I where τ t I τ 1 1 x cosτx dx I τ 1 1 x sinτx dx. Now by 7 8 we obtain I 1 t cos t t 18 sin F :; 1:; : 1/ 1; ; 1 t /6 / : ; F :;1 1:;1 : 1/ 1; 1; 1 t /6 : ; /;.

11 CONSTRUCTION OF ORTHONORMAL WAVELETS 9 Now we find the associated mother wavelet ψ : 17 ψ t e iw/ ˆψ we itw dw 1 { 1 9w 6w + costw dw w + 1w 7costw dw w 8 + w 9w 8 8 6w 1costw dw } +8 costw dw. e iw/ ˆψ wcostw dw We denote these integrals by J 1 J J J 4 respectively start with J 1 : J 1 w costw dw γ cos t γ + dγ [ ] cos t cost/ sin t +. t t Likewise 8 J 4 w 8 costw dw γ cos t γ + 8 dγ [ ] cos8t/ cost t sin t + t J 4/ 9 w 4 costw dw / 9 γ cos t γ + 4 dγ 1/ 4 1 x cos t τx dx where τ t/ I I are given by 7 8. Thus J 1 4t cos F :; : 1/ 1; ; 1:; 1 t /6 / : ; ; + t 4t 18 sin F :;1 1:;1 [ cos 4t ] I +sin4t I : 1/ 1; 1; 1 t /6 : ; /;.

12 94 AHMED I. ZAYED As for J wehave J w 4 costw dw 1/ 1 x cos τx+ 4t dx { cos 4t I τ sin 4t } I τ 4t cos F :; : 1/ 1; 1:; 1 t /9 / : ; ; t sin 4t 9 F :;1 : 1/ 1; 1; 1:;1 1 t /9. : ; /; Substituting J i i1 4 back into equation 17 yields 1. The wavelets given in the above theorem are entire functions of exponential type are given explicitly by Taylor series with known coefficients. This makes it easy to calculate them to any desired degree of accuracy. References [1] I. Daubechies Ten Lectures on Wavelets SIAM Publications Soc. Indust. Appl. Math. Philadelphia 199. MR 9e:445 [] A.ErdelyiW.MagnusF.OberhettingerF.TricomiHigher Transcendental Functions Vol. 1 McGraw-Hill New York 195. MR 15:419i [] H. Exton Multiple Hypergeometric Functions And Applications John Wiley & Sons New York MR 54:1699 [4] I. Gradshteyn I. Ryzhik Tables of Integrals Series Products Academic Press New York MR :595 [5] E. Hernez G. Weiss A First Course on Wavelets CRC Press Boca Raton Florida MR 97i:415 [6] H. Srivastava H. Manocha A Treatise on Generating functions John Wiley & Sons New York MR 85m:16 [7] W.H. Young On infinite integrals involving a generalization of the sine cosine functions Quart. J. Math. Vol pp [8] G. Walter Translation dilation invariance in orthogonal wavelets Appl. Comp. Harmonic Anal. Vol pp MR 96b:44 [9] A. Zayed G. Walter Wavelets in Closed Forms in Wavelet Transforms Timefrequency Signal Analysis Appl. Numer. Harmon. Anal. Birkhäuser Boston MA 1 pp MR c:461 Department of Mathematical Sciences DePaul University Chicago Illinois address: azayed@condor.depaul.edu