Orthonormal Wavelet System in ${\ell }^2({\mathbb{Z}}_N^2)$

Abstract

Using the group theoretic approach based on the set of digits, we first investigate a finite collection of functions in ${\ell }^2({\mathbb{Z}}_N^2)$ that satisfies some localization properties in a region of the time-frequency plane. The digits are associated with an invertible (expansive/non-expansive) matrix having integer entries. Next, we study and characterize an orthonormal wavelet system for ${\ell }^2({\mathbb{Z}}_N^2)$. In addition, some results connecting the uncertainty principle with functions that generate the orthonormal wavelet system having time-frequency localization properties are obtained.

arXiv comment14 pages