The functional-analytic properties of the limit \(q\)-Bernstein operator
type: Item
instance of: scholarly article
title: The functional-analytic properties of the limit \(q\)-Bernstein operator
zbMATH Open document ID: 1268.46022
DOI: 10.1155/2012/280314
author: Sofiya Ostrovska
published in: Journal of Function Spaces and Applications
publication date: 2013-02-05T00:00:00Z
review text: Summary: The limit \(q\)-Bernstein operator \(B_q\), \(0 < q < 1\), emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution. The latter is used in the \(q\)-boson theory to describe the energy distribution in a \(q\)-analogue of the coherent state. Lately, the limit \(q\)-Bernstein operator has been widely under scrutiny, and it has been shown that \(B_q\) is a positive shape-preserving linear operator on \(C[0, 1]\) with \(||B_q|| = 1\). Its approximation properties, probabilistic interpretation, eigenstructure, and impact on the smoothness of a function have been examined. In this paper, the functional-analytic properties of \(B_q\) are studied. Our main result states that there exists an infinite-dimensional subspace \(M\) of \(C[0, 1]\) such that the restriction \(B_q|_M\) is an isomorphic embedding. Also, we show that each such subspace \(M\) contains an isomorphic copy of the Banach space \(c_0\).
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 6134537
zbMATH Keywords: \(q\)-Bernstein operator
zbMATH Keywords: isomorphic embeddings
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/280314
OpenAlex ID: W2014677899
Wikidata QID: Q58908174
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description: scientific article; zbMATH DE number 6134537