Construction of Wavelet Analysis in the Space of Discrete Splines Using Zak Transform

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 Bsplines 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 l2.