Bernstein Operators for Extended Chebyshev Systems

TL;DR

This paper investigates conditions under which Bernstein operators can be constructed for extended Chebyshev systems, ensuring they reproduce specific functions and preserve shape properties within these function spaces.

Contribution

It establishes criteria for the existence of Bernstein operators that reproduce given functions in extended Chebyshev spaces, extending classical approximation theory.

Findings

Conditions for the existence of Bernstein operators in extended Chebyshev systems

Construction of operators that reproduce specific functions

Application of these operators to shape-preserving approximation

Abstract

Let $U_{n}\subset C^{n}[ a,b] $ be an extended Chebyshev space of dimension $n+1$. Suppose that $f_{0}\in U_{n}$ is strictly positive and $% f_{1}\in U_{n}$ has the property that $f_{1}/f_{0}$ is strictly increasing. We search for conditions ensuring the existence of points $% t_{0},...,t_{n}\in [ a,b] $ and positive coefficients $\alpha_{0},...,\alpha_{n}$ such that for all $f\in C[ a,b]$, the operator $B_{n}:C[ a,b] \to U_{n}$ defined by $% B_{n}f=\sum_{k=0}^{n}f(t_{k}) \alpha_{k}p_{n,k}$ satisfies $% B_{n}f_{0}=f_{0}$ and $B_{n}f_{1}=f_{1}.$ Here it is assumed that $% p_{n,k},k=0,...,n$, is a Bernstein basis, defined by the property that each $% p_{n,k}$ has a zero of order $k$ at $a$ and a zero of order $n-k$ at $b.$