Bernstein operators for exponential polynomials

  1. Aldaz, J.M. 1
  2. Kounchev, O. 2
  3. Render, H. 1
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  2. 2 Institute of Mathematics and Informatics
    info

    Institute of Mathematics and Informatics

    Sofía, Bulgaria

    ROR https://ror.org/01b5dy719

Revista:
Constructive Approximation

ISSN: 0176-4276

Ano de publicación: 2009

Volume: 29

Número: 3

Páxinas: 345-367

Tipo: Artigo

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DOI: 10.1007/S00365-008-9010-6 SCOPUS: 2-s2.0-62949224161 WoS: WOS:000264262000004 arXiv: 0805.1618v1 GOOGLE SCHOLAR lock_openAcceso aberto editor

Outras publicacións en: Constructive Approximation

Repositorio institucional: lockAcceso aberto Editor

Resumo

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 Bnf:= ∑nk=0αkf(tkP n,k(x) satisfies Bneλjx, 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]. © 2008 Springer Science+Business Media, LLC.