On some properties of modulation spaces as Banach algebras

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-Lévy theorem for $M_s^{p,1}({\mathbf{R}}^n)$, and clarify the sets of spectral synthesis for $M_s^{p,1}({\mathbf{R}}^n)$ by using the ``ideal theory for Segal algebras’’ developed in Reiter [30].The inclusion relationship between the modulation space $M_0^{p,1}(\mathbf{R})$ and the Fourier Segal algebra $\mathcal{F}{A}_p(\mathbf{R})$ is also determined.