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BIT Numerical Mathematics

, Volume 55, Issue 3, pp 797–821 | Cite as

Superconvergent \(C^1\) cubic spline quasi-interpolants on Powell-Sabin partitions

  • Driss SbibihEmail author
  • Abdelhafid Serghini
  • Ahmed Tijini
  • Ahmed Zidna
Article

Abstract

In this paper we introduce a B-spline representation of the cubic Hermite Powell-Sabin interpolant of any polynomial or any piecewise polynomial, over Powell-Sabin partitions of class at least \(C^1\), in terms of their polar forms. We use this B-spline representation for constructing several superconvergent discrete cubic spline quasi-interpolants which approximate a function \(f\) better than the superconvergent quadratic ones developed in one of our recent published papers. The new results presented in this work are an improvement and a generalization of those studied recently in the literature. We also illustrate by numerical examples that global errors and cubature rules based on these cubic Powell-Sabin spline quasi-interpolants are positively affected by the superconvergence phenomenon.

Keywords

Polar forms Quasi-interpolation splines Powell-Sabin partitions 

Mathematics Subject Classification

41A05 65D05 65D15 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Driss Sbibih
    • 1
    Email author
  • Abdelhafid Serghini
    • 2
  • Ahmed Tijini
    • 1
  • Ahmed Zidna
    • 3
  1. 1.Laboratoire MATSI, FSO-EST, URAC05Université Mohammed 1erOujdaMaroc
  2. 2.Laboratoire MATSI, EST, URAC05Université Mohammed 1erOujdaMaroc
  3. 3.LITAUniversité de LorraineMetzFrance

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