Gabor-like systems in $L^2({\bf R}^d)$ and extensions to wavelets

TL;DR

This paper presents a method to construct orthonormal bases in multi-dimensional L^2 spaces from Gabor bases in one dimension, extending to frames and wavelets using unitary operators.

Contribution

It introduces a novel approach to generate multi-dimensional bases from one-dimensional Gabor bases via unitary operators, including extensions to frames and wavelets.

Findings

Constructed orthonormal bases in L^2(R^d) from Gabor bases in L^2(R)

Extended the method to frames and wavelets

Discussed numerous examples of the construction

Abstract

In this paper we show how to construct a certain class of orthonormal bases in $L^2({\bf R}^d)$ starting from one or more Gabor orthonormal bases in $L^2({\bf R})$. Each such basis can be obtained acting on a single function $\Psi(\underline x)\in L^2({\bf R}^d)$ with a set of unitary operators which operate as translation and modulation operators {\em in suitable variables}. The same procedure is also extended to frames and wavelets. Many examples are discussed.