Approximation of functions by means of a new generalized Bernstein operator
type: Item
instance of: scholarly article
title: Approximation of functions by means of a new generalized Bernstein operator
zbMATH Open document ID: 0544.41022
DOI: 10.1007/BF02575593
author: Dimitrie D. Stancu
published in: Calcolo
publication date: 1983-01-01T00:00:00Z
review text: The paper presents some new contributions of the author to the theory of constructive approximation theory. First, one describes an urn model for deducing a discrete probability distribution which permits to construct a new positive linear operator of Bernstein type: \(L_{m,r}^{\alpha,\beta}\), depending on a non-negative integer parameter r and on two real parameters \(\alpha\) and \(\beta\), such that 0\(\leq \alpha \leq \beta\), and which is defined, for any \(f:[0,1]\to {\mathbb{R}},\) by \[ (L_{m,r}^{\alpha,\beta}f)(x):= \sum^{m-r}_{k=0} p_{m-r,k}(x)[(1-x)f(k+\alpha /m+\beta)+xf(k+r+\alpha /m+\beta)], \] where \(p_{m-r,k}(x):= \binom{m-r}{k} x^ k(1-x)^{m-r-k}\). The approximation properties of this operator are investigated in detail. A special attention is accorded to the case of the operator \(L_{m,r}=L^{o,o}_{m,r}\). One establishes several representations for the remainder of the approximation formula of the function f by means of this operator and one evaluates the orders of approximation, in terms of modulus of continuity of f, or of f'. The point spectrum of the operator \(L_{m,r}\) is determined and the quadrature formula which can be obtained by using this operator is given. Further new results and extensions of these operators can be found in the following recent papers of the author: 1) A note on a multiparameter Bernstein-type approximating operator, Mathematica (Cluj) 26(49), 153-157 (1984); 2) Generalized Bernstein operators, ''Babeş-Bolyai'' Univ., Faculty of Math., Research Seminaries, Preprint Nr. 6, 1984, Itinerant Seminar on functional equations, approximation and convexity, 185-192, Cluj-Napoca (1984); 3) Probabilistic approach to a class of generalized Bernstein approximating operators, Math.-Rev. Anal. Numér. Théor. Approximation. Anal. Numér. Théor. Approximation 14(1985) (in print).
Mathematics Subject Classification ID: 41A36
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A40
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 60C05
Mathematics Subject Classification ID: 65D32
zbMATH DE Number: 3865904
zbMATH Keywords: discrete probability distribution
zbMATH Keywords: remainder
zbMATH Keywords: orders of approximation
zbMATH Keywords: quadrature formula
MaRDI profile type: Publication
cites work: Q3941809
cites work: Eine Verallgemeinerung der Bernsteinschen Operatoren
cites work: Q5543516
cites work: Sulle formule di quadratura
cites work: Q3262494
cites work: On the degree of approximation by linear positive operators
cites work: On approximation by linear positive operators
cites work: Q4758872
cites work: Q3266531
cites work: Restausdrücke bei Interpolations- und Quadraturformeln durch bestimmte Integrale
cites work: Evaluation of the Remainder Term in Approximation Formulas by Bernstein Polynomials
cites work: The Remainder of Certain Linear Approximation Formulas in Two Variables
cites work: Q5605336
cites work: Q5579142
full work available at URL: https://doi.org/10.1007/bf02575593
OpenAlex ID: W2079879539
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description: scientific article; zbMATH DE number 3865904