Bernstein operators for exponential polynomials

Details Bernstein operators for exponential polynomials Bernstein operators for exponential polynomials

2008-05-12

arxiv.org/abs/0805.1618v1

Constr. Approx. 29 (2009), no. 3, 345--367

Mathematics

Classical Analysis and ODEs

A very similar version is to appear in Constructive Approximation

Scientific paper

10.1007/s00365-008-9010-6

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}% \lambda_{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 $\lambda_{0}$ and $\lambda_{1}$ are real and distinct, our first main result states that there exist points $% a=t_{0}

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