LAUSR

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 $${\mathcal {M}}^F$$MF on $${\mathbb {R}}^{d}$$Rd, associated to translation–modulation invariant Banach spaces of ultradistributions F on $${\mathbb {R}}^{2d}$$R2d.