How large are the gaps in phase space?

TL;DR

This paper provides quantitative bounds on the size of gaps in phase space for wavelet and short-time Fourier transforms using Laguerre and Hermite functions, with bounds depending on the condition number.

Contribution

It introduces elementary, explicit bounds on phase space gaps that depend on the condition number, independent of the measure's structure.

Findings

Bounds depend on the condition number of sampling constants

Estimates are independent of the measure's structure

Proofs are elementary and rely on explicit formulas

Abstract

Given a sampling measure for the wavelet transform (resp. the short-time Fourier transform) with the wavelet (resp. window) being chosen from the family of Laguerre (resp. Hermite) functions, we provide quantitative upper bounds on the radius of any ball that does not intersect the support of the measure. The estimates depend on the condition number, i.e., the ratio of the sampling constants, but are independent of the structure of the measure. Our proofs are completely elementary and rely on explicit formulas for the respective transforms.