# Periodic distributions and periodic elements in modulation spaces

## TL;DR

This paper characterizes periodic elements in various function spaces, including modulation spaces, using Fourier coefficient estimates and short-time Fourier transforms, providing new duality results for these spaces.

## Contribution

It introduces a novel characterization of periodic elements in modulation and related spaces through Fourier and transform estimates, extending duality results.

## Key findings

- Duality between certain modulation spaces and their periodic elements
- Characterization of periodic elements via Fourier coefficients and transforms
- Extension of Bessel's identity to these function spaces

## Abstract

We characterize periodic elements in Gevrey classes, Gelfand-Shilov distribution spaces and modulation spaces, in terms of estimates of involved Fourier coefficients, and by estimates of their short-time Fourier transforms. If $q\in [1,\infty )$, $\omega$ is a suitable weight and $(\maclE _0^E)'$ is the set of all $E$-periodic elements, then we prove that the dual of $M^{\infty ,q}_{(\omega )}\cap (\maclE _0^E)'$ equals $M^{\infty ,q'}_{(1/\omega )}\cap (\maclE _0^E)'$ by suitable extensions of Bessel's identity.