Bernstein operators for extended Chebyshev systems |
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Authors: | JM Aldaz O Kounchev |
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Institution: | a Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco 28049, Madrid, Spain b Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria c School of Mathematical Sciences, University College Dublin, Dublin 4, Ireland |
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Abstract: | Let Un ⊂ Cna, b] be an extended Chebyshev space of dimension n + 1. Suppose that f0 ∈ Un is strictly positive and f1 ∈ Un has the property that f1/f0 is strictly increasing. We search for conditions ensuring the existence of points t0, …, tn ∈ a, b] and positive coefficients α0, …, αn such that for all f ∈ Ca, b], the operator Bn:Ca, b] → Un defined by satisfies Bnf0 = f0 and Bnf1 = f1. Here it is assumed that pn,k, k = 0, …, n, is a Bernstein basis, defined by the property that each pn,k has a zero of order k at a and a zero of order n − k at b. |
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Keywords: | Bernstein polynomial Bernstein operator Extended Chebyshev system Exponential polynomial |
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