A Note on the Generalized Bernstein Polynomials

Constructive Approximation, 1992

In this paper we give a complete expansion formula for Bernstein polynomials defined on a s-dimensional simplex. This expansion for a smooth function f represents the Bernstein polynomial B n (f ) as a combination of derivatives of f plus an error term of order O(n -s ).

2004

We introduce polynomials B n i (x; ω|q), depending on two parameters q and ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonnière, as well as q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating B n i (x; ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented.

Applied Mathematics and Computation, 2011

In this paper, we consider the modified q-Bernstein polynomials for functions of several variables on q-Volkenborn integral and investigate some new interesting properties of these polynomials related to q-Stirling numbers, Hermite polynomials and Carlitz’s type q-Bernoulli numbers.

Mediterranean Journal of Mathematics, 2019

We consider some connections between the classical sequence of Bernstein polynomials and the Taylor expansion at the point 0 of a C ∞ function f defined on a convex open subset Ω ⊂ R d containing the d-dimensional simplex S d of R d. Under general assumptions, we obtain that the sequence of Bernstein polynomials converges to the Taylor expansion and hence to the function f together with derivatives of every order not only on S d but also on the whole Ω. This result yields extrapolation properties of the classical Bernstein operators and their derivatives. An extension of the Voronovskaja's formula is also stated.

Journal of Mathematical Analysis and Applications, 2009

In this paper, we introduce the generalized q-Bernstein polynomials based on the q-integers and we study approximation properties of these operators. In special case, we obtain Stancu operators or Phillips polynomials.

arXiv preprint arXiv:1111.4849, 2011

Abstract: The aim of this paper is to give a new approach to modified q-Bernstein polynomials for functions of several variables. By using these polynomials, recurrence formulas and some new interesting identities related to the second Stirling numbers and generalized Bernoulli polynomials are derived. Moreover, the generating function and interpolation function of these polynomials of several variables and also the derivatives of these polynomials and their generating function are given.

Applied Mathematics & Information Sciences, 2017

The aim of this paper is to give a new approach to modified q-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling numbers and generalized Bernoulli polynomials are derived. Moreover, the generating function, interpolation function of these polynomials of several variables and also the derivatives of these polynomials and their generating function are given. Finally, we get new interesting identities of modified q-Bernoulli numbers and q-Euler numbers applying p-adic q-integral representation on Z p and p-adic fermionic q-invariant integral on Z p , respectively, to the inverse of q-Bernstein polynomials.

Let Bm(f ) be the Bernstein polynomial of degree m. The generalized Bernstein polynomials

Advances in Difference Equations, 2015

In this study we examine generating functions for the Bernstein type polynomials given in (Simsek in Fixed Point Theory Appl. 2013:80, 2013). We expand these generating functions using the parameters u and v. By applying these generating functions, we obtain some functional equations and partial differential equations. In addition, using these equations, we derive several identities and relations related to these polynomials. Finally, numerical values of these polynomials for selected cases are demonstrated with their plots.

Czechoslovak Mathematical Journal, 2012

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