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 $$ has the property that $f_{1}/f_{0}$ is strictly increasing. We search for conditions ensuring the existence of points $$ and positive coefficients $α_{0},...,α_{n}$ 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 $b.$

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