An invariant analytic orthonormalization procedure with applications

TL;DR

This paper presents an invariant orthonormalization method applied to Gabor frames, enabling the construction of mutually orthogonal functions with potential applications in signal processing.

Contribution

It introduces a novel invariant orthonormalization procedure and demonstrates its application to Gabor frames, enhancing the construction of orthogonal functions in functional analysis.

Findings

Constructed orthogonal functions for Gabor frames.

Analyzed the role of the associated lattice in orthonormalization.

Extended the procedure to other examples beyond Gabor frames.

Abstract

We apply the orthonormalization procedure previously introduced by two of us and adopted in connection with coherent states to Gabor frames and other examples. For instance, for Gabor frames we show how to construct $g(x)\in L^2(\Bbb{R})$ in such a way the functions $g_{\underline n}(x)=e^{ian_1x}g(x+an_2)$, $\underline n\in\Bbb{Z}^2$ and $a$ some positive real number, are mutually orthogonal. We discuss in some details the role of the lattice naturally associated to the procedure in this analysis.