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-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.