Coorbit theory of warped time-frequency systems in $\mathbb{R}^d$

TL;DR

This paper extends warped time-frequency systems to higher dimensions, establishing conditions for their localization, and demonstrates how to discretize these systems to analyze sparsity in coorbit spaces.

Contribution

It generalizes warped time-frequency analysis to multiple dimensions and provides conditions for localization and discretization, enabling sparse representations in coorbit spaces.

Findings

Higher-dimensional warped time-frequency systems retain key properties.

Conditions for Gramian localization are established.

Discrete frames can be constructed via sampling, linking sparsity to coorbit space membership.

Abstract

Warped time-frequency systems have recently been introduced as a class of structured continuous frames for functions on the real line. Herein, we generalize this framework to the setting of functions of arbitrary dimensionality. After showing that the basic properties of warped time-frequency representations carry over to higher dimensions, we determine conditions on the warping function which guarantee that the associated Gramian is well-localized, so that associated families of coorbit spaces can be constructed. We then show that discrete Banach frame decompositions for these coorbit spaces can be obtained by sampling the continuous warped time-frequency systems. In particular, this implies that sparsity of a given function $f$ in the discrete warped time-frequency dictionary is equivalent to membership of $f$ in the coorbit space. We put special emphasis on the case of radial warping functions, for which the relevant assumptions simplify considerably.