Shape preserving properties of univariate Lototsky–Bernstein operators

Abstract

The main goal of this paper is to study shape preserving properties of univariate Lototsky–Bernstein operators Ln(f) based on Lototsky–Bernstein basis functions. The Lototsky–Bernstein basis functions bn,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 pi(x), where pi(0)=0 and pi(1)=1 for 1≤i≤n. These operators Ln(f) are positive linear operators that preserve constant functions, and a non-constant function γnp(x). If all the pi(x) are strictly increasing and strictly convex, then γnp(x) is strictly increasing and strictly convex as well. Iterates LnM(f) of Ln(f) are also considered. It is shown that LnM(f) converges to f(0)+(f(1)−f(0))γnp(x) as M→∞. Like classical Bernstein operators, these Lototsky–Bernstein operators enjoy many traditional shape preserving properties. For every (1,γnp(x))-convex function f∈C[0,1], we have Ln(f;x)≥f(x); and by invoking the total positivity of the system {bn,k(x)}0≤k≤n, we show that if f is (1,γnp(x))-convex, then Ln(f;x) is also (1,γnp(x))-convex. Finally we show that if all the pi(x) are monomial functions, then for every (1,γn+1p(x))-convex function f, Ln(f;x)≥Ln+1(f;x) if and only if p1(x)=⋯=pn(x)=x.