Gabor frames and τ-pseudodifferential operators on locally compact abelian groups

Abstract

This paper presents a comprehensive framework for time-frequency analysis on locally compact abelian groups. We introduce and study generalized versions of fundamental time-frequency objects, specifically the cross τ-ambiguity function and the cross τ-Wigner distribution. Building on these definitions, we define a class of τ-pseudodifferential operators via a duality approach, establishing that the modulation spaces Mp,rω(G) serve as their natural functional setting. A significant portion of our work is dedicated to the theory of Gabor frames on quasi-lattices. We prove that under mild conditions on the window function, the associated coefficient and reconstruction operators are bounded on modulation spaces, leading to unconditional convergence of the resulting frame expansions. This provides a powerful machinery for the phase-space analysis of pseudodifferential operators in a general group-theoretic context, unifying and extending classical results from Euclidean space to the setting of locally compact abelian groups.