Global and Partial Fourier Series for Denjoy-Carleman Classes on the Torus

TL;DR

This paper develops Fourier analysis techniques for Denjoy-Carleman classes of ultradifferentiable functions on the torus, enabling generalizations of classical results like the Greenfield-Wallach Theorem.

Contribution

It introduces Fourier analysis methods tailored for Denjoy-Carleman classes, expanding the scope of harmonic analysis in ultradifferentiable function spaces.

Findings

Established Fourier series representations for Denjoy-Carleman classes.

Generalized the Greenfield-Wallach Theorem using these Fourier analysis techniques.

Extended classical harmonic analysis results to ultradifferentiable function spaces.

Abstract

In this work, we develop Fourier Analysis for a family of classes of ultradifferentiable functions of Romieu type on the torus, usually known as Denjoy-Carleman classes. Then we are able to apply our results in order to generalize some results whose proofs rely almost exclusively on Fourier Series, such as the Greenfield-Wallach Theorem.