Kernel identities and vectorial regularisation
type: Item
instance of: scholarly article
title: Kernel identities and vectorial regularisation
zbMATH Open document ID: 1402.46030
author: Christian Bargetz
author: Norbert Ortner
published in: Monatshefte für Mathematik
publication date: 2018-06-27T00:00:00Z
review text: The aim of this article is to present a new method to prove some known and new kernel identities. More precisely, let \(\dot{{\mathcal B}}\) denote the space of smooth functions with derivatives vanishing at infinity and let \(\dot{{\mathcal B}}'\) be the closure of compactly supported distributions in the class of bounded distributions. In Section, 2 the authors obtain the identities \[ \dot{{\mathcal B}}_{xy} = \dot{{\mathcal B}}_{x}\widehat{\otimes}_\varepsilon \dot{{\mathcal B}}_{y},\;\;\dot{{\mathcal B}}'_{xy} = \dot{{\mathcal B}}'_{x}\widehat{\otimes}_\varepsilon \dot{{\mathcal B}}'_{y}, \] (Propositions 3 and 4), while in Section 3, the authors prove \[ {\mathcal D}_{L_1,xy} = {\mathcal D}_{L_1,x}\widehat{\otimes}_\pi {\mathcal D}_{L_1,y}, \;\;{\mathcal D}'_{L_1,xy} = {\mathcal D}'_{L_1,x}\widehat{\otimes}_\pi {\mathcal D}'_{L_1,y}, \] (Propositions 9 and 10). All the identities are obtained after applying appropriate vectorial regularization properties obtained in Propositions 1 and 7. For instance, the identity \({\mathcal D}_{L_1,xy} = {\mathcal D}_{L_1,x}\widehat{\otimes}_\pi {\mathcal D}_{L_1,y}\) follows from \[ K(x,z)\in {\mathcal D}_{L_1,x}\widehat{\otimes}_\pi E_z \Leftrightarrow K(x-y,z)\in \left({\mathcal E}_y\widehat{\otimes}L_x^1\right)\widehat{\otimes}_\pi E_z. \]
reviewed by: Antonio Galbis
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 46A32
zbMATH DE Number: 6894843
zbMATH Keywords: vector-valued distributions
zbMATH Keywords: regularization of distributions
zbMATH Keywords: convolution of distributions
zbMATH Keywords: completed topological tensor products
Wikidata QID: Q59607760
MaRDI profile type: Publication
full work available at URL: s00605-017-1076-4
OpenAlex ID: W3103216874
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DOI: 10.1007/S00605-017-1076-4
description: scientific article; zbMATH DE number 6894843