The approximation properties of generalized Bernstein polynomials of two variables
type: Item
instance of: scholarly article
title: The approximation properties of generalized Bernstein polynomials of two variables
zbMATH Open document ID: 1063.41015
published in: Applied Mathematics and Computation
publication date: 2004-11-23T00:00:00Z
review text: The authors generalize the 2-dimensional Bernstein polynomials along the lines of Stancu's generalization of of the ordinary Bernstein polynomials, namely, for parameters \(0\leq\alpha_k\leq\beta_k\), \(k=1,2\), and a continuous function \(f\) defined on the unit square, they denote \[ B^{\alpha_k,\beta_k}_{n,m}(f;x,y):=\sum_{i=0}^n\sum_{j=0}^mf\left(\frac{i+\alpha_1} {n+\beta_1},\frac{j+\alpha_2}{m+\beta_2}\right)\binom ni\binom mjx^i(1-x)^{n-i}y^j(1-y)^{m-j}. \] The authors prove that their generalized Bernstein polynomials converge uniformly in \([0,1]\times[0,1]\) to \(f\) and obtain some estimates on the degree of approximation involving the modulus of continuity of \(f\). The results are straightforward, moreover, considering that similar results are known for the ordinary 2-dimensional Bernstein polynomials, all that is needed is the trivial estimate of the distance between the point \(\left(\frac{i+\alpha_1} {n+\beta_1},\frac{j+\alpha_2}{m+\beta_2}\right)\) and the point \(\left(\frac in,\frac jm\right)\). The latter tends to zero uniformly in \(0\leq i\leq n\) and \(0\leq j\leq m\), as \(n,m\to\infty\). In fact, all one needs is to observe that \[ \frac{i+\alpha_1}{n+\beta_1}-\frac in=\frac{n\alpha_1+i\beta_1}{n(n+\beta_1)}\leq\frac{2\beta_1}{n+\beta_1}, \] and similarly for the other pair. The reviewer sees no point in this generalization and no benefit in the figures, tables, and obvious computer programs in the article.
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 2118022
zbMATH Keywords: 2-dimensional Bernstein polynomials
zbMATH Keywords: degree of approximation
author: İbrahim Büyükyazıcı
author: Ertan Ibikli
reviewed by: D. Leviatan
MaRDI profile type: Publication
full work available at URL: j.amc.2003.07.023
OpenAlex ID: W2079250736
cites work: Q3254589
cites work: Q5819609
cites work: Application of Bernstein polynomials for smooth estimation of a distribution and density function
cites work: The Bernstein Polynomials and Finite Differences
DOI: 10.1016/J.AMC.2003.07.023
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description: scientific article; zbMATH DE number 2118022