On some properties of modulation spaces as Banach algebras

TL;DR

This paper explores the properties of modulation spaces as Banach algebras, proving the Wiener-Lévy theorem and analyzing spectral synthesis, thereby advancing the understanding of their algebraic and spectral structures.

Contribution

It establishes the Wiener-Lévy theorem for modulation spaces and clarifies their spectral synthesis sets using ideal theory, providing new insights into their algebraic properties.

Findings

Proved Wiener-Lévy theorem for modulation spaces

Clarified spectral synthesis sets for modulation spaces

Determined inclusion relations between modulation and Fourier Segal algebras

Abstract

In this paper, we give some properties of the modulation spaces $M_s^{p,1}({\mathbf R}^n)$ as commutative Banach algebras. In particular, we show the Wiener-L\'evy theorem for $M^{p,1}_s({\mathbf R}^n)$, and clarify the sets of spectral synthesis for $M^{p,1}_s ({\mathbf R}^n)$ by using the ``ideal theory for Segal algebras'' developed in Reiter [30].The inclusion relationship between the modulation space $M^{p,1}_0 ({\mathbf R})$ and the Fourier Segal algebra ${\mathcal F}\hspace{-0.08cm}A_p({\mathbf R})$ is also determined.