A de Casteljau algorithm for generalized Bernstein polynomials

1997 ◽  
Vol 37 (1) ◽  
pp. 232-236 ◽  
Author(s):  
George M. Phillips
Keyword(s):  
Bernstein Polynomials ◽  
De Casteljau Algorithm ◽  
Generalized Bernstein Polynomials

scholarly journals A de Casteljau Algorithm for -Bernstein-Stancu Polynomials

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Grzegorz Nowak
Keyword(s):  
Bernstein Polynomials ◽  
Classical Case ◽  
Geometric Progression ◽  
De Casteljau Algorithm ◽  
Stancu Operators

This paper is concerned with a generalization of the -Bernstein polynomials and Stancu operators, where the function is evaluated at intervals which are in geometric progression. It is shown that these polynomials can be generated by a de Casteljau algorithm, which is a generalization of that relating to the classical case and -Bernstein case.

Generalized Bernstein Polynomials

2004 ◽  
Vol 44 (1) ◽  
pp. 63-78 ◽  
Author(s):  
Stanisław Lewanowicz ◽  
Paweł Woźny
Keyword(s):  
Bernstein Polynomials ◽  
Generalized Bernstein Polynomials

scholarly journals A generalization of the Bernstein polynomials

1999 ◽  
Vol 42 (2) ◽  
pp. 403-413 ◽  
Author(s):  
Haul Oruç ◽  
George M. Phillips ◽  
Philip J. Davis
Keyword(s):  
Bernstein Polynomials ◽  
Classical Case ◽  
Geometric Progression ◽  
Generalized Bernstein Polynomials

This paper is concerned with a generalization of the classical Bernstein polynomials where the function is evaluated at intervals which are in geometric progression. It is shown that, when the function is convex, the generalized Bernstein polynomials Bn are monotonic in n, as in the classical case.

scholarly journals Convergence of Generalized Bernstein Polynomials

2002 ◽  
Vol 116 (1) ◽  
pp. 100-112 ◽  
Author(s):  
Alexander Il'inskii ◽  
Sofiya Ostrovska
Keyword(s):  
Bernstein Polynomials ◽  
Generalized Bernstein Polynomials

Integration of Jacobi and Weighted Bernstein Polynomials Using Bases Transformations

2007 ◽  
Vol 7 (3) ◽  
pp. 221-226 ◽  
Author(s):  
Abedallah Rababah
Keyword(s):  
Jacobi Polynomials ◽  
Bernstein Polynomials ◽  
Definite Integral ◽  
De Casteljau Algorithm ◽  
Bernstein Bases

AbstractThis paper presents methods to compute integrals of the Jacobi polynomials by the representation in terms of the Bernstein — B´ezier basis. We do this because the integration of the Bernstein — B´ezier form simply corresponds to applying the de Casteljau algorithm in an easy way. Formulas for the definite integral of the weighted Bernstein polynomials are also presented. Bases transformations are used. In this paper, the methods of integration enable us to gain from the properties of the Jacobi and Bernstein bases.

ON GENERALIZED BERNSTEIN POLYNOMIALS

1996 ◽  
pp. 263-269 ◽  
Author(s):  
GEORGE M PHILLIPS
Keyword(s):  
Bernstein Polynomials ◽  
Generalized Bernstein Polynomials

Summability of generalized Bernstein polynomials. I

1955 ◽  
Vol 22 (4) ◽  
pp. 617-623 ◽  
Author(s):  
P. L. Butzer
Keyword(s):  
Bernstein Polynomials ◽  
Generalized Bernstein Polynomials

The r-monotonicity of generalized Bernstein polynomials

2012 ◽  
Vol 55 (3) ◽  
pp. 797-807
Author(s):  
Laiyi Zhu ◽  
Zhiyong Huang
Keyword(s):  
Continuous Function ◽  
Bernstein Polynomials ◽  
Sufficient Condition ◽  
Generalized Bernstein Polynomials

AbstractLet f ∊ C[0, 1] and let the Bn(f, q; x) be generalized Bernstein polynomials based on the q-integers that were introduced by Phillips. We prove that if f is r-monotone, then Bn(f, q; x) is r-monotone, generalizing well-known results when q = 1 and the results when r = 1 and r = 2 by Goodman et al. We also prove a sufficient condition for a continuous function to be r-monotone.

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