Wavelets and polylogarithms of negative integer order

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1 Wavelets and polylogarithms of negative integer order T.E.Krenkel, E.T.Krenkel, K.O.Egiazarian, and J.T.Astola Moscow Technical University for Communications and Informatics, Department of Computer Science, Russia, Signal Processing Laboratory, Tampere University of Technology, Tampere, Finland, Abstract Wavelets are objects on the real line which are stable under the action of an affine group (dilations and translations). In the Fourier domain they can be designed with the polylogarithms l s (x) = e 2πinx /n s for x R/Z and Re(s) >, n= satisfying (for every positive integer m) the following relation (Kubert identity)[]: m φ(mx) = m s j=0 φ(x + m j) ( s ). Using the periodic polylogarithms of the negative integer order s = k =, 2, 3,... we demonstrate how orthonormalized wavelets of Battle-Lemarie ([2] and [3]) can be designed. Introduction Recently [4] we have demonstrated that using the polylogarithms [5] of the negative integer order, we can design the subfamily of even Battle-Lemarie wavelets. This example leads us to a general method of reduction of the wavelet design problems to the problems of number theory. 2 Periodic zeta function The function l s (x) is also known as a periodic polylogarithm function or periodic zeta function [5]. Evidently, l s (x) satisfies the Kubert identities ( s ) as well as the identity l s (x)/ x = 2πil s. ()

2 If Re(s) >, we can write l s (x) = cos(2πnx)/n s + i sin(2πnx)/n s, where the summands on the right side are the even and odd parts of l s (x). If s is real, these can be identified with the real and imaginary parts of l s (x). The polylogarithm may be defined for Re(s) > 0 and z < as the function (Jonquiere function): L s (z) = z t s Γ(s) 0 e t dt, s > 0. z Derivatives of the polylogarithm are specified as z d dz L s(z) = L s (z), (z d dz )m L s (z) = L s m (z). From the first equation we have Hence, by the second equation, L 0 (z) = z z, L (z) = L (z) = ln( z). z ( z) 2, L z(z + ) 2 = ( z) 3, L 3 = z(z2 + 4z + ) ( z) In general, L m (z) is easy to compute for integer m : L m (z) = ( z) m+ m a m,k z k, for m > 0, k= where the coefficients ( Eulerian numbers) can be obtained by the recurrence equation: a m,k = (m + k)a m,k + ka m,k. The numerator in the equation of the polylogarithm is equal to a corresponding Euler-Frobenius polynomial. It is evident that for negative integer values of the parameter s, the functions L s (z) are rational, with rational coefficients, holomorphic in z except for a pole at z =. It is also obvious that l s (x) = L s (e 2πix ) for x R/Z, x 0, and for all complex s. For the function l 0 (x) = e 2πix /( e 2πix ), a brief computation shows that l 0 (x) = ( + i cot(πx))/2. Differentiating this expression, we obtain corresponding formulas for the periodic polylogarithmic functions of negative integer order l (x), l 2 (x),.... Note, in particular, that l s (x) is either an odd or an even function respectively, as s is odd or even, for every negative integer s. 2

3 3 Zhamalov-Weiss polynomials and Battle- Lemarie wavelets The first five of the Zhamalov polynomials are as follows [4]: A (w) = w, A 2 (w) = 6w 2 4w, A 3 (w) = 20w 3 20w 2 + 6w, A 4 (w) = 5040w w w 2 64w, A 5 (w) = w w w w w. where ω is frequency and is a substitute for x in the previous text and It is evident from the above that w = sin 2 πω. l 2M (ω)/2 = Γ(2M) π 2M A M(w), M =, 2, 3,.... (2) Remark. Zhamalov was the first to introduce in the early 70-ies even polynomials in the process of finding a fundamental solution of a polyharmonic equation on R/Z by inverting the polyharmonic operator in the Fourier domain. He also derived a reccurent formula for the calculation of even polynomials [6]. Remark 2. It is also noteworthy that Weil in his book [7] also introduced even and odd functions in the following form: and ζ(2m, ω) + ζ(2m, ω) = {Ψ(2M, ω) + Ψ(2M, ω)} (2M )! ζ(2m, ω) ζ(2m, ω) = { Ψ(2M 2, ω) + Ψ(2M 2, ω)}, (2M)! where Ψ(t, ω) is the t-th polygamma function, which is t-th derivative of the digamma function Ψ(ω). It is known that (ω + 2nπ) 2 = 4sin 2 (ω/2). Differentiating twice both sides of this equality we get for every ω R (ω + 2nπ) 4 = 6sin 4 (ω/2) { 2 3 sin2 (ω/2)} which is clearly the second even Zhamalov polynomial in variable w = sin 2 (ω/2). 3

4 The technique of design of the even Battle-Lemarie wavelets using the radicals of Zhamalov polynomials is described in [4]. If one wants to design the even and odd Battle-Lemarie wavelets in a unified manner, the technique described by Hernandez and Weiss [8] [see Ch.4.2] should be used. Let us assume the validity for the case t 2 of the following equality: (ω + 2nπ) t = P t (ω/2) (2 sin(ω/2)) t, where P t (ω) is a trigonometric polynomial. Differentiating both sides of this equality we obtain (ω + 2nπ) t+ = P t (ω/2) (2 sin(ω/2)) t+, where P t (ω) is a Weiss trigonometric polynomial (i. e., a polynomial in sin ω and cos ω). Put P, then the recurrent formula for Weiss polynomials looks like: P t (ω) = (cos ω)p t (ω) /t(sin ω)p t (ω). The first five of the Weiss polynomials are as follows: P (ω) =, P 2 (ω) = cos ω, P 3 (ω) = 2 3 sin2 ω = cos(2ω), P 4 (ω) = 3 cos3 ω cos ω, P 5 (ω) = 30 cos2 (2ω) cos(2ω) In the Fourier domain we have the following equations for the design of the scaling functions with t = 2M + : ( ) M+ sin(ω/2) ˆϕ M ω/2 if M is odd, P2M+ (ω/2) (ω) = ( ) M+ (3) e iω/2 sin(ω/2) ω/2 if M is even, P2M+ (ω/2) low-pass filters where m M 0 (ω) = and wavelets ˆψ M (2ω) = ϕˆ M (2ω) ϕˆ M (ω) = eiα M (ω) (cos(ω/2)) M+ P 2M+ (ω/2) P 2M+ (ω), (4) { α M (ω) = 0 if M is odd, ω/2 if M is even, e iω (sin(ω/2)) 2M+2 (ω/2) M+ P2M+ (ω/2+π/2) P 2M+ (ω)p 2M+ (ω/2) ie iω (sin(ω/2)) 2M+2 (ω/2) M+ P2M+ (ω/2+π/2) P 2M+ (ω)p 2M+ (ω/2) if M is odd, if M is even. As a result of the above, a technique for the design of the Battle-Lemarie wavelets in the Fourier domain using polylogarithms of negative integer order was obtained. 4 (5)

5 References [] D. Kubert, The universal ordinary distribution, Bull. Soc. math. France 07 (979), [2] G. Battle, A block spin construction of ondolettes. Part I: Lemarie functions, Commun. Math. Phys. 0 (987), [3] P.-G. Lemarie, Ondolettes a localisation exponentielles, J. Math. pures et appl. 67 (988), [4] T. Krenkel, E. Krenkel, K. Egiazarian and J. Astola, Sobolev-Zhamalov wavelets: Battle-Lemarie wavelets revisited I, (unpublished report). [5] L. Lewin, Polylogarithms and Associated Functions, North Holland, 98. [6] S. L. Sobolev. The Theory of Cubature Formulas, Kluwer, 997. [7] A. Weil, Elliptic functions according to Eisenstein and Kronecker, Springer, New York, 976. [8] E. Hernandez, G. Weiss, A first course on wavelets, CRC Press, Boca Raton,

Compactly Supported Tight Frames Associated with Refinable Functions 1. Communicated by Guido L. Weiss Received July 27, 1999

Applied and Computational Harmonic Analysis 8, 93 319 (000) doi:10.1006/acha.000.0301, available online at http://www.idealibrary.com on LETTER TO THE EDITOR Compactly Supported Tight Frames Associated