Translation-modulation invariant Banach spaces of ultradistributions

Abstract

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 MF on Rd , associated to translation–modulation invariant Banach spaces of ultradistributions F on R2d .