Pseudodifferential operators on time-frequency invariant Banach spaces and applications to Gabor Frames

TL;DR

This paper investigates pseudodifferential operators with periodic symbols and their continuity and invertibility on time-frequency invariant Banach spaces, leading to new conditions for Gabor frame existence on L^2 spaces.

Contribution

It introduces new results on the continuity and invertibility of Gabor operators with periodic symbols on Banach spaces, and provides conditions for Gabor frame existence for general lattices.

Findings

Gabor operators with periodic symbols are continuous on certain Banach spaces.

Invertibility conditions for these operators are established.

Sufficient conditions for Gabor frame existence on L^2 spaces are derived.

Abstract

Starting from the study of pseudodifferential operators with completely periodic symbols, we obtain results of continuity and invertibility of a class of Gabor operators on time-frequency invariant Banach spaces. As an applications we find sufficient conditions for the existence of Gabor frames on the space of square integrable functions, associated to a general lattice.