Geometric series of Bernstein operators revisited

Journal of Mathematical Analysis and Applications, 2012

We give an affirmative answer to a conjecture of G.T. Tachev concerning the moments of the Bernstein operators.

Mediterranean Journal of Mathematics

We study generalizations of the classical Bernstein operators on the polynomial spaces P n [a, b], where instead of fixing 1 and x, we reproduce exactly 1 and a polynomial f 1 , strictly increasing on [a, b]. We prove that for sufficiently large n, there always exist generalized Bernstein operators fixing 1 and f 1. These operators are defined by non-decreasing sequences of nodes precisely when f ′ 1 > 0 on (a, b), but even if f ′ 1 vanishes somewhere inside (a, b), they converge to the identity.

2008

In this article we want to determinate a recursive formula for Bernstein polynomials associated to the functions ep(x) = xp, p ∈ N, and an expresion for the central moments of the Bernstein polinomyals. 2000 Mathematics Subject Classification. 41A10; 41A63.

Results in Mathematics, 2009

Kelisky and Rivlin have proved that the iterates of the Bernstein operator (of fixed order) converge to L, the operator of linear interpolation at the endpoints of the interval [0, 1]. In this paper we provide a large class of (not necessarily positive) linear bounded operators T on C[0, 1] for which the iterates T n converge towards L in the operator norm. The proof uses methods from the spectral theory of linear operators.

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

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.

SERIES III - MATEMATICS, INFORMATICS, PHYSICS

In this paper we investigate certain properties of a class of generalized Bernstein type operators

2016

By using an univalent and analytic function τ in a suitable open disk centered in origin, we attach to analytic functions f, the complex Bernstein-type operators of the form B_{n,q}^{τ}(f)=B_{n,q}(f∘τ⁻¹)∘τ , where B_{n,q} denote the classical complex q-Bernstein polynomials, q≥1. The new complex operators satisfy the same quantitative estimates as B_{n,q}. As applications, for two concrete choices of τ, we construct complex rational functions and complex trigonometric polynomials which approximate f with a geometric rate.

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.

Mathematische Zeitschrift, 1967

Journal of Approximation Theory, 2000

The Bernstein operator B n reproduces the linear polynomials, which are therefore eigenfunctions corresponding to the eigenvalue 1. We determine the rest of the eigenstructure of B n . Its eigenvalues are (n) k := n! (n ? k)! 1 n k ; k = 0; 1; : : :; n; and the corresponding monic eigenfunctions p (n) k are polynomials of degree k, which have k simple zeros in 0; 1]. By using an explicit formula, it is shown that p (n) k converges as n ! 1 to a polynomial related to a Jacobi polynomial. Similarly, the dual functionals to p (n) k converge as n ! 1 to measures that we identify. This diagonal form of the Bernstein operator and its limit, the identity (Weierstrass density theorem), is applied to a number of questions. These include the convergence of iterates of the Bernstein operator, and why Lagrange interpolation (at n + 1 equally spaced points) fails to converge for all continuous functions whilst the Bernstein approximants do. We also give the eigenstructure of the Kantorovich operator. Previously, the only member of the Bernstein family for which the eigenfunctions were known explicitly was the Bernstein{Durrmeyer operator, which is self adjoint.

2011

In this paper, derivatives of the product of Bernstein polynomials of the same and different degrees are obtained. Also a recurrence formula for those polynomials together with some new properties are given.

American Mathematical Monthly, 2009

Computers & Mathematics with Applications, 2011

In this paper we present the sequence of linear Bernstein-type operators defined for f ∈ C [0, 1] by B n (f • τ −1) • τ , B n being the classical Bernstein operators and τ being any function that is continuously differentiable ∞ times on [0, 1], such that τ (0) = 0, τ (1) = 1 and τ ′ (x) > 0 for x ∈ [0, 1]. We investigate its shape preserving and convergence properties, as well as its asymptotic behavior and saturation. Moreover, these operators and others of King type are compared with each other and with B n. We present as an interesting byproduct sequences of positive linear operators of polynomial type with nice geometric shape preserving properties, which converge to the identity, which in a certain sense improve B n in approximating a number of increasing functions, and which, apart from the constant functions, fix suitable polynomials of a prescribed degree. The notion of convexity with respect to τ plays an important role.

Numerical Algorithms, 2013

The Bernstein operators of second kind were introduced by Paolo Soardi in 1990, in terms of a random walk on a certain hypergroup. They have the same relation with Chebyshev polynomials of second kind as the classical Bernstein operators have with Chebyshev polynomials of first kind. In this paper we describe a de Casteljau type algorithm for these operators. Keywords Bernstein operators of second kind • Chebyshev polynomials • Random walks on hypergroups Mathematics Subject Classifications 41A36 • 20N20 • 20P05