Multivariate generalized Bernstein polynomials: Identities for orthogonal polynomials of two variables
type: Item
instance of: scholarly article
title: Multivariate generalized Bernstein polynomials: Identities for orthogonal polynomials of two variables
zbMATH Open document ID: 1183.33021
author: Stanisław Lewanowicz
author: Paweł Woźny
author: Iván Area
author: Eduardo Godoy
published in: Numerical Algorithms
publication date: 2009-04-29T00:00:00Z
review text: The authors define polynomials \(B_\mathbf{k}^n(\mathbf{x}; \omega | q)\), which they call multivariate generalized Bernstein polynomials, depending on the two parameters \(q\) and \(\omega\). These polynomials inherit many special cases and identities can be retrieved from limit forms. These polynomials extend the multivariate classical and discrete Bernstein polynomials [see \textit{G. G. Lorentz}, Bernstein polynomials. 2nd ed. New York, NY: Chelsea Publishing. (1986; Zbl 0989.41504)] as well as the univariate \(q\)-Bernstein polynomials introduced by \textit{G. M. Phillips} [Ann. Numer. Math. 4, No. 1--4, 511--518 (1997; Zbl 0881.41008)]. In analogy to the classical multivariate Bernstein polynomials, basic properties of the new polynomials are given, such as recurrence relations with respect to \(\mathbf{k}\) and \(n\), \(q\)-differentiation rules, the de Casteljau algorithm, the partition of unity property and a symmetry property. Furthermore, connections to bivariate big \(q\)-Jacobi polynomials are made. Finally, relations between bivariate \(q\)-Bernstein and little \(q\)-Jacobi polynomials and, respectively, between bivariate discrete Bernstein and Hahn polynomials are given. An application of the polynomials in CAGD is outlined in the Introduction.
reviewed by: Thomas Stoll
Mathematics Subject Classification ID: 33C50
Mathematics Subject Classification ID: 33D50
Mathematics Subject Classification ID: 65D17
zbMATH DE Number: 5549480
zbMATH Keywords: bivariate big \(q\)-Jacobi polynomials
zbMATH Keywords: bivariate \(q\)-Hahn polynomials
zbMATH Keywords: bivariate Hahn polynomials
zbMATH Keywords: connection relations
MaRDI profile type: Publication
full work available at URL: s11075-008-9168-9
OpenAlex ID: W2073703495
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DOI: 10.1007/S11075-008-9168-9
description: scientific article; zbMATH DE number 5549480