Further study of modulation spaces as Banach algebras

TL;DR

This paper investigates the spectral synthesis properties of modulation spaces that are Banach algebras, extending previous work and deriving variants of Wiener-Lévy theorems for these spaces.

Contribution

It extends spectral synthesis analysis to a broader class of modulation spaces forming Banach algebras, including new variants of Wiener-Lévy theorems.

Findings

Spectral synthesis results for modulation spaces as Banach algebras

Extension of previous case q=1 to general q

Derivation of Wiener-Lévy type theorems for these spaces

Abstract

This paper discusses spectral synthesis for those modulation spaces $M^{p,q}_s({\mathbf R}^n)$ which form Banach algebras under pointwise multiplication. An important argument will be the ``ideal theory for Segal algebras'' by H. Reiter [15]. This paper is a continuation of our paper [5] where the case $q=1$ is treated. As a by-product we obtain a variant of Wiener-L\'evy theorem for $M^{p,q}_s({\mathbf R}^n)$ and Fourier-Wermer algebras ${\mathcal F}L^q_s({\mathbf R}^n)$.