An answer to a conjecture on Bernstein operators (original) (raw)
Related papers
On multiplicativity of the Bernstein operator
Computers & Mathematics with Applications, 2011
For continuous functions f and g, we prove that the Bernstein operator B n is multiplicative for all n ≥ 1 and all x ∈ 2[0, 1] if and only if at least one of the functions f and g is a constant function. Some other variants of multiplicativity are also considered.
An extremal property of Bernstein operators
Journal of Approximation Theory, 2007
We establish a strong version of a known extremal property of Bernstein operators, as well as several characterizations of a related specific class of positive polynomial operators.
A problem of I. Ra\c{s}a on Bernstein polynomials and convex functions
arXiv: Classical Analysis and ODEs, 2016
We present an elementary proof of a conjecture by I. Rasa which is an inequality involving Bernstein basis polynomials and convex functions. It was affirmed in positive very recently by the use of stochastic convex orderings. Moreover, we derive the corresponding results for Mirakyan-Favard-Szasz operators and Baskakov operators.
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.
On (p,q)-analogue of Bernstein Operators (Revised)
arXiv (Cornell University), 2015
In the present article, we have given a corrigendum to our paper "On (p, q)-analogue of Bernstein operators" published in Applied Mathematics and Computation 266 (2015) 874-882.
The Eigenstructure of the Bernstein Operator
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.