Bernstein operators for exponential polynomials

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Title: Bernstein operators for exponential polynomials
Authors: Aldaz, J. M.
Kounchev, Ognyan
Render, Hermann
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Date: Apr-2009
Online since: 2014-03-28T09:30:16Z
Abstract: 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 {| Im λ j | : j = 0 ,...,n } , 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 B n f := n X k =0 α k f ( t k ) p n,k ( x ) 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 ∈ C [ a,b ].
Type of material: Journal Article
Publisher: Springer
Journal: Constructive Approximation
Volume: 29
Issue: 3
Start page: 345
End page: 367
Copyright (published version): 2009 Springer
Keywords: Bernstein polynomialBernstein operatorExtended chebyshev systemExponential polynomial
DOI: 10.1007/s00365-008-9010-6
Language: en
Status of Item: Peer reviewed
Appears in Collections:Mathematics and Statistics Research Collection

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