On the invertibility of "rectangular" bi-infinite matrices and applications in time--frequency analysis

TL;DR

This paper extends classical invertibility results of finite matrices to bi-infinite matrices and applies these findings to time-frequency analysis, specifically in the context of p-frames and operator identifiability.

Contribution

It generalizes invertibility properties to bi-infinite matrices and applies these to derive density and identifiability results in time-frequency analysis.

Findings

Density results for p-frames of time-frequency molecules

Identifiability results for operators with bandlimited Kohn--Nirenberg symbols

Generalization of invertibility properties to bi-infinite matrices

Abstract

Finite dimensional matrices having more columns than rows have no left inverses while those having more rows than columns have no right inverses. We give generalizations of these simple facts to bi--infinite matrices and use those to obtain density results for $p$--frames of time--frequency molecules in modulation spaces and identifiability results for operators with bandlimited Kohn--Nirenberg symbols.