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

TL;DR

This paper investigates the construction and characterization of orthonormal wavelet systems in finite-dimensional discrete spaces using group theory, linking localization properties and uncertainty principles.

Contribution

It introduces a group-theoretic approach to characterize orthonormal wavelet systems in finite discrete spaces and explores their localization and uncertainty properties.

Findings

Characterization of orthonormal wavelet systems in $\, ext{ell}^2({ ext{Z}}_N^2)$

Connection between localization properties and the uncertainty principle

Conditions for functions to generate wavelet systems with desired properties

Abstract

Using the group theoretic approach based on the set of digits, we first investigate a finite collection of functions in $\ell^2 ({\mathbb{Z}}^2_N)$ 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 $\ell^2 ({\mathbb{Z}}^2_N)$. In addition, some results connecting the uncertainty principle with functions that generate the orthonormal wavelet system having time-frequency localization properties are obtained.