Bernstein Operators for Extended Chebyshev Systems

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 has the property that $f_1/f_0$ is strictly increasing. We search for conditions ensuring the existence of points 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 satisfies and $B_n{f}_1=f_1.$ Here it is assumed that , is a Bernstein basis, defined by the property that each has a zero of order $k$ at $a$ and a zero of order $n-k$ at $b.$

17 pages

References in corpus (2)

• Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces
• Bernstein operators for exponential polynomials

Cited by in corpus (3)

• Generalized Bernstein operators defined by increasing nodes
• Generalized Bernstein operators on the classical polynomial spaces
• Convergence of rational Bernstein operators