Bernstein Operators for Extended Chebyshev Systems

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 $% f_{1}\in U_{n}$hasthepropertythat$f_{1}/f_{0}$ is strictly increasing. We search for conditions ensuring the existence of points $$ and positive coefficients ${\alpha }_0,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_n$ such that for all $f\in C[ a,b]$,theoperator$B_{n}:C[ a,b] \to U_{n}$definedby$% B_{n}f=\sum_{k=0}^{n}f(t_{k}) \alpha_{k}p_{n,k}$satisfies$% B_{n}f_{0}=f_{0}$and$B_{n}f_{1}=f_{1}.$Hereitisassumedthat$% p_{n,k},k=0,…,n$, is a Bernstein basis, defined by the property that each $$ has a zero of order $k$ at $a$ and a zero of order $n-k$ at $b\mathrm{.}$