We introduce and study new distribution spaces, the test function space $\mathcal{D}_E$ and its strong dual $\mathcal{D}'_{E'_{\ast}}$. These spaces generalize the Schwartz spaces $\mathcal{D}_{L^{q}}$, $\mathcal{D}'_{L^{p}}$, $\mathcal{B}'$ and their weighted versions. The construction of our new distribution spaces is based on the analysis of a suitable translation-invariant Banach space of distributions $E$, which turns out to be a convolution module over a Beurling algebra $L^{1}_{\omega}$. The Banach space $E'_{\ast}$ stands for $L_{\check{\omega}}^1\ast E'$. We also study convolution and multiplicative products on $\mathcal{D}'_{E'_{\ast}}$
We introduce and study a number of new spaces of ultradifferentiable functions and ultradistributions and we apply our results to the study of the convolution of ultradistributions. The spaces of convolutors O-C'*(R-d) for tempered ultradistributions are analyzed via the duality with respect to the test function spaces O-C*(R-d) introduced in this article. We also study ultradistribution spaces associated to translation-invariant Banach spaces of tempered ultradistributions and use their properties to provide a full characterization of the general convolution of Roumieu ultradistributions via the space of integrable ultradistributions. We show that the convolution of two Roumieu ultra distributions T, S is an element of D'({Mp}) (R-d) exists if and only if (phi * S)T is an element of D-L1'({Mp}) (R-d) for every phi is an element of D-{Mp} (R-d)
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 2d .
We study boundary values of holomorphic functions in translation-invariant distribution spaces of type D-E' . New edge of the wedge theorems are obtained. The results are then applied to represent D-E' as a quotient space of holomorphic functions. We also give representations of elements D-E' of via the heat kernel method. Our results cover as particular instances the cases of boundary values, analytic representations and heat kernel representations in the context D-LP' , B' of the Schwartz spaces and their weighted versions
We define ultradistributional wave front sets with respect to translationmodulation invariant Banach spaces of ultradistributions having solid Fourier image. The main result is their characterisation by the short-time Fourier transform. 2010 Mathematics Subject Classification. 35A18, 42B10, 46F05.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.