Sampling time-frequency localized functions and constructing localized time-frequency frames

TL;DR

This paper investigates functions with localized time-frequency content, providing approximation methods using eigenfunctions and local Gabor systems to construct concentrated time-frequency dictionaries.

Contribution

It introduces approximation inequalities and a framework for building localized time-frequency dictionaries based on eigenfunctions and Gabor systems.

Findings

Approximation inequalities for localized functions

Construction of localized time-frequency dictionaries

Use of eigenfunctions and Gabor systems for concentration

Abstract

We study functions whose time-frequency content are concentrated in a compact region in phase space using time-frequency localization operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators, as well as a local Gabor system covering the region of interest. These would allow the construction of modified time-frequency dictionaries concentrated in the region.