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 ${\mathcal{M}}^F$ on ${\mathbb{R}}^d$, associated to translation-modulation invariant Banach spaces of ultradistributions $F$ on ${\mathbb{R}}^{2d}$.

19 pages