### Bernstein operators for exponential polynomials

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Title: | Bernstein operators for exponential polynomials |

Authors: | Aldaz, J. M. Kounchev, Ognyan Render, Hermann |

Permanent link: | http://hdl.handle.net/10197/5498 |

Date: | Apr-2009 |

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 |

Copyright (published version): | 2009 Springer |

Keywords: | Bernstein polynomial;Bernstein operator;Extended chebyshev system;Exponential 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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