Dense sets in spaces of integrable functions
type: Item
instance of: scholarly article
title: Dense sets in spaces of integrable functions
zbMATH Open document ID: 0878.46018
author: Paweł Rolicz
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-12-11T00:00:00Z
review text: The author establishes some theorems concerning approximation of \(L^{p}\)-functions by 'polynomials of many variables' \(g\in L(\mathcal F)\) in which the role of variables play functions from a given set \(\mathcal F\) (i.e., \(L(\mathcal F) := \{g=\sum_{\alpha }^{}c_{\alpha }f_{1}^{\alpha _{1}}\cdot f_{2}^{\alpha _{2}}\cdot\dots\cdot f_{m}^{\alpha _{m}}; f_{i}\in \mathcal F\}\), where \(m\) are nonnegative integers, \(\alpha := (\alpha_{1},\dots,\alpha _{m})\) are multiindices, and \(c_{\alpha }\) are complex constants which differ from zero for a finite number of multiindices only). The paper defines the space \(L^{p}\) as a space of functions integrable with \(p-\)th power on a Hausdorff space \(T,\) being the countable union of compact subsets, with respect to a finite and regular Borel measure [and hence it represents a generalization of the paper of the same author, J. Math. Anal. Appl. 181, No. 2, 505-523 (1994; Zbl 0801.46031)]). Various sufficient conditions on \(\mathcal F\) are established which guarantee the density of the above-mentioned 'polynomials': the functions from \(\mathcal F\) are supposed to separate the points, some continuity properties are requested, and the set \(\mathcal F\) contains together with any function \(f\) its conjugate, too. To this aim, the author uses a slightly modified version of the Stone-Weierstrass theorem: he shows that the `Stone-Weierstrass approximation polynomials' from \(\mathcal F\) which tend to the function \(f\) can be chosen in such a way that their absolute values are uniformly bounded, which makes possible -- with the help of the Luzin theorem -- to use Lebesgue's dominated convergence theorem.
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 28A25
Mathematics Subject Classification ID: 41A30
zbMATH DE Number: 786116
zbMATH Keywords: approximation of \(L^ p\)-functions
zbMATH Keywords: Stone-Weierstrass theorem
zbMATH Keywords: Luzin theorem
zbMATH Keywords: Lebesgue dominated convergence theorem
reviewed by: Pavel Doktor
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1157
OpenAlex ID: W1988220875
DOI: 10.1006/JMAA.1995.1157
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description: scientific article; zbMATH DE number 786116