# The Zak transform and Wiener estimates on Gelfand-Shilov and modulation spaces with applications to operator theory

## TL;DR

This paper characterizes Gelfand-Shilov and modulation spaces via Zak transform estimates, explores quasi-periodic functions, and provides conditions for operators to be conjugated by Zak transforms, with implications for operator theory.

## Contribution

It introduces new characterizations of function spaces using Zak transform estimates and establishes criteria for operator conjugation in this context.

## Key findings

- Characterization of Gelfand-Shilov and modulation spaces via Zak transform estimates
- Necessary and sufficient conditions for operator conjugation by Zak transform
- Applications to quasi-periodic functions and distributions

## Abstract

We characterize Gelfand-Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these result for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by the Zak transform.