A remark on dilation-and-modulation frames for $L^2(\mathbb{R}_+)$

TL;DR

This paper demonstrates that rationally sampled dilation-and-modulation systems in $L^2( plus)$ are unitarily equivalent to multi-window Gabor systems, enabling the transfer of Gabor analysis results to these systems.

Contribution

It establishes a unitary equivalence between dilation-and-modulation systems and multi-window Gabor systems, facilitating the application of Gabor analysis to dilation-and-modulation frameworks.

Findings

Rationally sampled dilation-and-modulation systems are unitarily equivalent to multi-window Gabor systems.

Frame properties from Gabor analysis can be applied to dilation-and-modulation systems.

This equivalence simplifies the analysis of dilation-and-modulation frames.

Abstract

We show that every rationally sampled dilation-and-modulation system is unitarily equivalent with a multi-window Gabor system. As a consequence, frame theoretical results from Gabor analysis can be directly transferred to dilation-and-modulation systems.