# A Paley-Wiener theorem in extended Gevrey regularity

## TL;DR

This paper develops a Paley-Wiener theorem for extended Gevrey classes by analyzing associated functions with specific sequences and applying Lambert W function asymptotics, impacting the understanding of wave front sets.

## Contribution

It introduces a new associated function for a specific sequence and proves a Paley-Wiener theorem in extended Gevrey regularity, linking asymptotic analysis with Fourier transform properties.

## Key findings

- Derived sharp asymptotic estimates using Lambert W function
- Proved a Paley-Wiener theorem for extended Gevrey classes
- Analyzed properties of wave front sets in this context

## Abstract

In this paper we introduce appropriate associated function to the sequence $M_p=p^{\t p^{\s}}$, $p\in \N$, $\t>0$, $\s>1$, and derive its sharp asymptotic estimates in terms of the Lambert $W$ function. These estimates are used to prove a Paley-Wiener type theorem for compactly supported functions from extended Gevrey classes. As an application, we discuss properties of the corresponding wave front sets.