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Lacunary Pál-Type Interpolation and Over-Convergence

Lacunary Pál-Type Interpolation and Over-Convergence Hermite-Birkhoff interpolation and Pál-type interpolation have been receiving much attention over the years. Also during the previous 15 years the subject of interpolation in non-uniformly distributed nodes has been looked into. There are, however, not many examples known where lacunary problems (in which the orders of the derivatives for which data are given, are non-consecutive) are regular.Here lacunary Pál-type interpolation is looked into: the (0, m1, …, mq) interpolation problem on the zeros of \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$z^{n}- \alpha_{0}^{n},z^{n}- \alpha_{1}^{n},\cdots z^{n}- \alpha_{q}^{n}$\end{document} with \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$0< \alpha_{0}< \alpha_{1}< \cdots< \alpha_{q}$\end{document} will be shown to be regular for n sufficiently large.Moreover, for a power series with radius of convergence \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\rho >\alpha_{q}$\end{document} the phenomenon of over-convergence of the difference between the interpolating polynomial of the full power series and that of the partial sums of length (q+1)nℓ will be exhibited as n → ∞ where ℓ ≥ 1 is an arbitrary integer. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Lacunary Pál-Type Interpolation and Over-Convergence

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Publisher
Springer Journals
Copyright
Copyright © Heldermann  Verlag 2003
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/bf03321040
Publisher site
See Article on Publisher Site

Abstract

Hermite-Birkhoff interpolation and Pál-type interpolation have been receiving much attention over the years. Also during the previous 15 years the subject of interpolation in non-uniformly distributed nodes has been looked into. There are, however, not many examples known where lacunary problems (in which the orders of the derivatives for which data are given, are non-consecutive) are regular.Here lacunary Pál-type interpolation is looked into: the (0, m1, …, mq) interpolation problem on the zeros of \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$z^{n}- \alpha_{0}^{n},z^{n}- \alpha_{1}^{n},\cdots z^{n}- \alpha_{q}^{n}$\end{document} with \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$0< \alpha_{0}< \alpha_{1}< \cdots< \alpha_{q}$\end{document} will be shown to be regular for n sufficiently large.Moreover, for a power series with radius of convergence \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\rho >\alpha_{q}$\end{document} the phenomenon of over-convergence of the difference between the interpolating polynomial of the full power series and that of the partial sums of length (q+1)nℓ will be exhibited as n → ∞ where ℓ ≥ 1 is an arbitrary integer.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Mar 1, 2004

Keywords: Pál-type interpolation; regularity; over-convergence; 41A05

References