An identity for multivariate Bernstein polynomials

doi:10.1016/j.cagd.2003.06.005

Computer Aided Geometric Design

Volume 20, Issues 8–9, November 2003, Pages 563-577

To the memory of Josef Hoschek

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Abstract

We prove an identity for multivariate Bernstein polynomials on simplices, which may be considered a pointwise orthogonality relation. Its integrated version provides a new representation for the polynomial dual basis of Bernstein polynomials. An identity for the reproducing kernel is used to define quasi-interpolants of arbitrary order.

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Cited by (12)

Multivariate Bernstein-Durrmeyer operators with arbitrary weight functions
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In this paper we introduce a class of Bernstein–Durrmeyer operators with respect to an arbitrary measure $ρ$ on the $d$ -dimensional simplex, and a class of more general polynomial integral operators with a kernel function involving the Bernstein basis polynomials. These operators generalize the well-known Bernstein–Durrmeyer operators with respect to Jacobi weights. We investigate properties of the new operators. In particular, we study the associated reproducing kernel Hilbert space and show that the Bernstein basis functions are orthogonal in the corresponding inner product. We discuss spectral properties of the operators. We make first steps in understanding convergence of the operators.
Complete asymptotic expansion for multivariate Bernstein-Durrmeyer operators and quasi-interpolants
2010, Journal of Approximation Theory
We establish complete asymptotic expansions in terms of certain differential operators for the Bernstein–Durrmeyer operators with Jacobi weights on the $d$ -dimensional simplices, and for their so-called natural quasi-interpolants. Our method extensively uses spectral properties of the involved operators. In particular, we prove new identities for the eigenvalues of the operators. We also show that the obtained asymptotic expansions can be differentiated term-by-term.
Studying Baskakov-Durrmeyer operators and quasi-interpolants via special functions
2007, Journal of Approximation Theory
We prove that the kernels of the Baskakov–Durrmeyer and the Szász–Mirakjan–Durrmeyer operators are completely monotonic functions. We establish a Bernstein type inequality for these operators and apply the results to the quasi-interpolants recently introduced by Abel. For the Baskakov–Durrmeyer quasi-interpolants, we give a representation as linear combinations of the original Baskakov–Durrmeyer operators and prove an estimate of Jackson–Favard type and a direct theorem in terms of an appropriate K-functional.
An identity for a general class of approximation operators
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We prove an identity for basis functions of a general family of positive linear operators. It covers as special cases the Bernstein, Szász–Mirakjan and Baskakov operators. A corollary of our result can be considered a pointwise orthogonality relation. The Bernstein case is the univariate case of a remarkable identity which recently was presented by Jetter and Stöckler. As an application we give a representation of a restricted dual basis and define a class of quasi-interpolants.
A new proof of an identity of Jetter and Stöckler for multivariate Bernstein polynomials
2006, Computer Aided Geometric Design
A simple proof of an elegant identity for multivariate Bernstein polynomials discovered recently by K. Jetter and J. Stöckler is given. This identity shows a pointwise orthogonality relation of multivariate Bernstein polynomials and implies a new representation for the dual basis of multivariate Bernstein polynomials. Our method is based on “generating functions”, which reveals a general structure of the identity.
Durrmeyer Operators and Their Natural Quasi-Interpolants
2006, Studies in Computational Mathematics
Citation Excerpt :
In the present paper, we have chosen the recursive definition (11) leading to a product representation for Uℓ,μ which was communicated to us by Michael Felten. The quasi-interpolants (13) were introduced in [17], for the unweighted case. The weighted case was considered in [18] and [4], where also the statements of Theorem 7 and Lemma 8 can be found.

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An identity for multivariate Bernstein polynomials

Abstract

J. Approx. Theory

J. Approx. Theory

J. Approx. Theory

J. Approx. Theory

Computer Aided Geometric Design