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.

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.

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.

scholarly journals Shape Preserving Properties forq-Bernstein-Stancu Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Yali Wang ◽  
Yinying Zhou
Keyword(s):  
Bernstein Polynomials ◽  
Shape Preserving ◽  
Positive Basis ◽  
Stancu Operators ◽  
Variation Diminishing ◽  
Totally Positive ◽  
Shape Preserving Properties ◽  
Monotonicity Preserving

We investigate shape preserving forq-Bernstein-Stancu polynomialsBnq,α(f;x)introduced by Nowak in 2009. Whenα=0,Bnq,α(f;x)reduces to the well-knownq-Bernstein polynomials introduced by Phillips in 1997; whenq=1,Bnq,α(f;x)reduces to Bernstein-Stancu polynomials introduced by Stancu in 1968; whenq=1,α=0, we obtain classical Bernstein polynomials. We prove that basicBnq,α(f;x)basis is a normalized totally positive basis on[0,1]andq-Bernstein-Stancu operators are variation-diminishing, monotonicity preserving and convexity preserving on[0,1].

scholarly journals Convexity and generalized Bernstein polynomials

1999 ◽  
Vol 42 (1) ◽  
pp. 179-190 ◽  
Author(s):  
Tim N. T. Goodman ◽  
Halil Oruç ◽  
George M. Phillips
Keyword(s):  
Positive Integer ◽  
Classical Result ◽  
Bernstein Polynomials ◽  
Total Positivity ◽  
Parameter Family ◽  
Geometric Progression ◽  
Generalized Bernstein Polynomials

In a recent generalization of the Bernstein polynomials, the approximated function f is evaluated at points spaced at intervals which are in geometric progression on [0, 1], instead of at equally spaced points. For each positive integer n, this replaces the single polynomial Bnf by a one-parameter family of polynomials , where 0 < q ≤ 1. This paper summarizes briefly the previously known results concerning these generalized Bernstein polynomials and gives new results concerning when f is a monomial. The main results of the paper are obtained by using the concept of total positivity. It is shown that if f is increasing then is increasing, and if f is convex then is convex, generalizing well known results when q = 1. It is also shown that if f is convex then, for any positive integer n This supplements the well known classical result that when f is convex.

scholarly journals A generalization of the Bernstein polynomials based on the q-integers

2000 ◽  
Vol 42 (1) ◽  
pp. 79-86 ◽  
Author(s):  
George M. Phillips
Keyword(s):  
Bernstein Polynomials ◽  
Geometric Progression ◽  
Equal Spacing

AbstractThis paper is concerned with a generalization of the Bernstein polynomials in which the approximated function is evaluated at points spaced in geometric progression instead of the equal spacing of the original polynomials.

A lambda calculus with naive substitution

1979 ◽  
Vol 28 (3) ◽  
pp. 269-282 ◽  
Author(s):  
John Staples
Keyword(s):  
Lambda Calculus ◽  
Classical Case ◽  
Syntactic Theory ◽  
Alternative Approach ◽  
New Formulation

AbstractAn alternative approach is proposed to the basic definitions of the lassical lambda calculus. A proof is sketched of the equivalence of the approach with the classical case. The new formulation simplifies some aspects of the syntactic theory of the lambda calculus. In particular it provides a justification for omitting in syntactic theory discussion of changes of bound variable.

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