On some properties of modulation spaces as Banach algebras

We give some properties of the modulation spaces $M_s^{p,1}(\mathbf R^n)$ as commutative Banach algebras. In particular, we prove the Wiener–Lévy 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 by Reiter. The inclusion relationship between the modulation space $M^{p,1}_0 (\mathbf R)$ and the Fourier Segal algebra $\mathcal FA_p(\mathbf R)$ is also determined.

Published in Open Access (under CC-BY license).