Pseudodifferential operators on time-frequency shift invariant ultradistributions with periodic symbols

Abstract In the present paper the authors study pseudodifferential operators with periodic symbols acting on time-frequency shift invariant spaces of generalized ultradistributions. Particular attention is devoted to symbols satisfying suitable Gelfand–Shilov type regularity conditions, which exhibit periodicity with respect to appropriate lattice structures. In this framework, we show how periodicity allows us to represent the corresponding operators in terms of time-frequency shifts. We then establish continuity and invertibility results on a broad class of Banach spaces of ultradistributions, extending previous results known in the tempered distribution setting. The analysis provides a natural functional framework for treating non-polynomial weights, including weights with exponential growth at infinity.