Bernstein operators for exponential polynomials
Let $L$ be a linear differential operator with constant coefficients of order $n$ and complex eigenvalues $λ_{0},...,λ_{n}$. Assume that the set $U_{n}$ of all solutions of the equation $Lf=0$ is closed under complex conjugation. If the length of the interval $[ a,b] $ is smaller than $π/M_{n}$, where $M_{n}:=\max \left\{| \text{Im}% λ_{j}| :j=0,...,n\right\} $, then there exists a basis $p_{n,k}$%, $k=0,...n$, of the space $U_{n}$ with the property that each $p_{n,k}$ has a zero of order $k$ at $a$ and a zero of order $n-k$ at $b,$ and each $% p_{n,k}$ is positive on the open interval $(a,b) .$ Under the additional assumption that $λ_{0}$ and $λ_{1}$ are real and distinct, our first main result states that there exist points $% a=t_{0}<t_{1}<...<t_{n}=b$ and positive numbers $α_{0},..,α_{n}$%, such that the operator \begin{equation*} B_{n}f:=\sum_{k=0}^{n}α_{k}f(t_{k}) p_{n,k}(x) \end{equation*} satisfies $B_{n}e^{λ_{j}x}=e^{λ_{j}x}$, for $j=0,1.$ The second main result gives a sufficient condition guaranteeing the uniform convergence of $B_{n}f$ to $f$ for each $f\in C[ a,b] $.