# Real Paley-Wiener theorems in spaces of ultradifferentiable functions

## TL;DR

This paper extends real Paley-Wiener theorems to ultradifferentiable function spaces and their transforms, providing new characterizations and analysis tools for functions with non-compact support.

## Contribution

It introduces novel Paley-Wiener theorems for ultradifferentiable classes and their transforms, including Gabor, Fourier, and Wigner, in multiple variables.

## Key findings

- Characterization of ultradifferentiable functions via Fourier and Wigner transforms
- Extension of Paley-Wiener theorems to non-compact support using polynomials
- New results for Gabor transform in the context of ultradifferentiable functions

## Abstract

We develop real Paley-Wiener theorems for classes ${\mathcal S}_\omega$ of ultradifferentiable functions and related $L^{p}$-spaces in the spirit of Bang and Andersen for the Schwartz class. We introduce results of this type for the so-called Gabor transform and give a full characterization in terms of Fourier and Wigner transforms for several variables of a Paley-Wiener theorem in this general setting, which is new in the literature. We also analyze this type of results when the support of the function is not compact using polynomials. Some examples are given.