Shape preserving properties of univariate Lototsky-Bernstein operators

JOURNAL OF APPROXIMATION THEORY

卷:224

Article

Xu, Xiao-Wei^1,2  Zeng, Xiao-Ming¹  Goldman, Ron² 

[1] Xiamen Univ, Sch Math Sci, Xiamen 361005, Peoples R China

[2] Rice Univ, Dept Comp Sci, Houston, TX 77005 USA

关键词: Lototsky-Bernstein operators;    Fixed point;    Iterates;    Shape preservihg;    Total positivity;   

DOI  :  10.1016/j.jat.2017.09.002

来源: Elsevier

【 摘 要 】

The main goal of this paper is to study shape preserving properties of univariate Lototsky-Bernstein operators L-n(f) based on Lototsky-Bernstein basis functions. The Lototsky-Bernstein basis functions b(n),(k)(x) (0 <= k <= n) of order n are constructed by replacing x in the ith factor of the generating function for the classical Bernstein basis functions of degree n by a continuous nondecreasing function p(i)(x), where p(i)(0) = 0 and p(i)(1) = 1 for 1 <= i <= n. These operators L-n(f) are positive linear operators that preserve constant functions, and a non-constant function gamma(p)(n)(x). If all the pi (x) are strictly increasing and strictly convex, then gamma(p)(n)(x) is strictly increasing and strictly convex as well. Iterates L-n(M)(f) of L-n(f) are also considered. It is shown that L-n(M)(f) converges to f (0) + (f (1) f(0))gamma(P)(n)(x) as M -> infinity. Like classical Bernstein operators, these Lototsky Bernstein operators enjoy many traditional shape preserving properties. For every (1, gamma(np) (x))-convex function f is an element of C[0, 1], we have L-n(f; x) >= f (x); and by invoking the total positivity of the system {b(n,k)(x)}0 <= k <= n, we show that if f is (1,gamma(p)(n): (x))-convex, then L-n(f; x) is also (1, gamma(p)(n))(x))-convex. Finally we show that if all the pi(x) are monomial functions, then for every (1, gamma(p)(n+1)(x))-convex function f, L-n(f; x) > Ln+1(f; x) if and only if p1 (x) = (...) = p(n)(x) = x. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free