Construction of wavelet analysis in the space of discrete splines using Zak transform Alexander B. Pevnyi * † and Valery A. Zheludev ‡ May 22, 2003 Abstract We consider equidistant discrete splines S(j ), j ∈ Z, which may grow as O(|j | s ) as |j |→∞. Such splines present a relevant tool for digital signal processing. The Zak transforms of B- splines yield the integral representation of discrete splines. We define the wavelet space as a weak orthogonal complement of the coarse-grid space in the fine-grid space. We establish the integral representation of the elements of the wavelet space. We define and characterize the wavelets whose shifts form bases of the wavelet space. By this means we design a wide library of bases for the space of discrete-time signals of power growth construct multiscale representation of this space. We provide formulas for processing such the signals by discrete spline wavelets. Constructed bases are at the same time the Riesz bases for the space l 2 . Keywords: Discrete spline, Zak transform, wavelets, signal processing. 1 Introduction In this paper we continue our study of discrete cardinal splines of power growth that was started in [18]. We construct the wavelet analysis in the space of such splines and suggest some algorithms for signal transforms by discrete spline wavelets. We stress that discrete splines and spline wavelets are defined on the set Z of integers and, as such, yield a natural tool for digital signal processing. In this area these splines and wavelets offer obvious advantages over the splines and wavelets of the continuous argument. The term cardinal spline means a spline with equidistant nodes kn where k ∈ Z, n is a fixed natural number. Cardinal splines of continuous argument of power growth were investigated in [21], [20] and [27]. In [27] the author presented an integral representation of such splines which allowed construction of wavelet analysis in the space of continuous splines of power growth [28]. In the present paper we develop a similar theory in the space of discrete splines of power growth using as a tool the Zak transforms ([25, 5]) of the discrete B–splines. Discrete splines first appeared in early seventies ([22]), but recently became the subject of extensive investigation ( [7] , [8, Chapter 6], [15], [16], [17], [18], [4]). We also mention a related work [19] which deals with wavelets of discrete argument. In [1] the authors operate with “quasi- discrete“ splines which are defined through sampling of the splines of the continuous argument. Using these splines the authors build the multiresolution analysis of the space l 2 . * Research supported by Russian Fund for Basic Research (grant 98-01-00196) † Dept. of Mathematics, Syktyvkar University, Syktyvkar, Russia, E-mail: pevnyi@ssu.edu.komi.com ‡ School of Computer Science, Tel Aviv University Tel Aviv 69978, Israel, E-mail: zhel@math.tau.ac.il 1