Approximation by \(q\)-Bernstein polynomials in the case \(q \rightarrow 1 +\)
type: Item
instance of: scholarly article
title: Approximation by \(q\)-Bernstein polynomials in the case \(q \rightarrow 1 +\)
zbMATH Open document ID: 1470.41007
DOI: 10.1155/2014/259491
author: Xuezhi Wu
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: Let \(B_{n, q}(f; x)\), \(q \in(0, \infty)\) be the \(q\)-Bernstein polynomials of a function \(f \in C [0,1]\). It has been known that, in general, the sequence \(\left(B_{n, q_n}(f)\right)\) with \(q_n \rightarrow 1 +\) is not an approximating sequence for \(f \in C [0,1]\), in contrast to the standard case \(q_n \rightarrow 1 -\). In this paper, we give the sufficient and necessary condition under which the sequence \(\left(B_{n, q_n}(f)\right)\) approximates \(f\) for any \(f \in C [0,1]\) in the case \(q_n > 1\). Based on this condition, we get that if \(1 < q_n < 1 + \ln 2 / n\) for sufficiently large \(n\), then \(\left(B_{n, q_n}(f)\right)\) approximates \(f\) for any \(f \in C [0,1]\). On the other hand, if \(\left(B_{n, q_n}(f)\right)\) can approximate \(f\) for any \(f \in C [0,1]\) in the case \(q_n > 1\), then the sequence \((q_n)\) satisfies \(\overline{\text{lim}}_{n \rightarrow \infty} n(q_n - 1) \leq \text{ln} 2\).
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A36
zbMATH DE Number: 7022024
Wikidata QID: Q59037091
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/259491
OpenAlex ID: W2034679619
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description: scientific article; zbMATH DE number 7022024