Explicit representations of spaces of smooth functions and distributions
type: Item
instance of: scholarly article
title: Explicit representations of spaces of smooth functions and distributions
zbMATH Open document ID: 1321.46025
author: Christian Bargetz
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-01-06T00:00:00Z
review text: In his recent paper [Math. Nachr. 287, No. 1, 10--22 (2014; Zbl 1301.46007)], the author constructed a common basis for \(C^\infty(\mathbb{R}^n)\) and \(D(\mathbb{R}^n)\) and several spaces of smooth functions in between (and, by duality, for the corresponding dual spaces). In the present paper, this basis is explicitly calculated. The main analytical result is the presentation of an explicit basis for the space \(\mathcal{E}_0:=\{f\in \mathcal{E}([0,1])\mid \forall k\in\mathbb{N}_0: f^{(k)}(1)=0\}\). This basis is obtained by means of a suitable transform of variables, showing that \(\mathcal{E}_0\) is isomorphic to \(\mathcal{S}_e\), the space of even Schwartz test functions, where even Hermite functions are a basis. Since \(\mathcal{E}_0\) is the building block in the sequence space representation for \(C^\infty(\mathbb{R}^n)\) and \(D(\mathbb{R}^n)\) in [the author, loc. cit.], the basis of \(\mathcal{E}_0\) leads to a common basis for \(C^\infty(\mathbb{R}^n)\) and \(D(\mathbb{R}^n)\) using topological tensor products. These are also used to treat the case of several variables.
reviewed by: Michael Langenbruch
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 46A35
zbMATH DE Number: 6383585
zbMATH Keywords: distributions
zbMATH Keywords: sequence spaces
zbMATH Keywords: Schauder basis
zbMATH Keywords: topological tensor products
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.12.009
OpenAlex ID: W1976883566
cites work: A sequence space representation of L. Schwartz' space \({{\mathcal{O}_{C}}}\)
cites work: Commutativity of the Valdivia-Vogt table of representations of function spaces
cites work: Q4828783
cites work: Einführung in die Theorie der lokalkonvexen Räume
cites work: Erratum to: Topological tensor products and nuclear spaces.
cites work: Q3918541
cites work: Tensor product of Schauder bases
cites work: Q4340900
cites work: Q488702
cites work: Q4722576
cites work: Espaces de fonctions différentiables à valeurs vectorielles
cites work: Théorie des distributions à valeurs vectorielles. II
cites work: Q5528072
cites work: Extension of C ∞ Functions Defined in a Half Space
cites work: Q4189654
cites work: Q3896047
cites work: A representation of the space (K).
cites work: Q3933455
cites work: A representation of the space \(O_ M\).
cites work: Q3951046
cites work: Q3669992
cites work: Unconditional and shrinking bases in locally convex spaces
DOI: 10.1016/J.JMAA.2014.12.009
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description: scientific article; zbMATH DE number 6383585