Translation–Modulation Invariant Banach Spaces of Ultradistributions

We introduce and study a new class of translation–modulation invariant Banach spaces of ultradistributions. These spaces show stability under Fourier transform and tensor products; furthermore, they have a natural Banach convolution module structure over a certain associated Beurling algebra, as well as a Banach multiplication module structure over an associated Wiener–Beurling algebra. We also investigate a new class of modulation spaces, the Banach spaces of ultradistributions M F on R d , associated to translation–modulation invariant Banach spaces of ultradistributions F on R 2 d .