On some properties of modulation spaces as Banach algebras

In this paper, we give some properties of the modulation spaces Mₛp,1( Rⁿ) as commutative Banach algebras. In particular, we show the Wiener-L\'evy theorem for Mp,1ₛ( Rⁿ), and clarify the sets of spectral synthesis for Mp,1ₛ ( Rⁿ) by using the ``ideal theory for Segal algebras'' developed in Reiter [30].The inclusion relationship between the modulation space Mp,1₀ ( R) and the Fourier Segal algebra F-0.08cmAₚ( R) is also determined.