Bernstein operators for exponential polynomials

Abstract

Let $L$ be a linear differential operator with constant coefficients of order $n$ and complex eigenvalues ${\lambda }_0,...,{\lambda }_n$. 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 $\pi /M_n$, 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 is positive on the open interval $(a,b).$ Under the additional assumption that ${\lambda }_0$ and ${\lambda }_1$ are real and distinct, our first main result states that there exist points and positive numbers ${\alpha }_0,..,{\alpha }_n$%, 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 $B_n{e}^{{\lambda }_{j}x}=e^{{\lambda }_{j}x}$, for $j=0,1.$ The second main result gives a sufficient condition guaranteeing the uniform convergence of $B_{n}f$ to $f$ for each $f\in C[a,b]$.

A very similar version is to appear in Constructive Approximation

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