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Error Bounds via Duchon’s Technique

  • Simon HubbertEmail author
  • Quôc Thông Lê Gia
  • Tanya M. Morton
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

The goal of this chapter is to investigate the SBF interpolation error when measured in the more general \(L_{p}-\)norms, where \(p \in [1,\infty ].\) To do this we once again revisit RBF theory and, in particular, we focus on Duchon’s strategy (see Definition  2.1) developed for \(D^{m}-\)splines in Euclidean space. The chapter itself falls into two parts. The first part carefully sets up spherical versions of the crucial results used in Duchon’s approach in \({\mathbb R}^{d}\) and in the second part we demonstrate how these results can be used, together with the variational framework for SBF interpolation, to provide the desired interpolation error estimates in the \(L_{p}\)-norms.

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Simon Hubbert
    • 1
    Email author
  • Quôc Thông Lê Gia
    • 2
  • Tanya M. Morton
    • 3
  1. 1.School of Economics, Mathematics and StatisticsBirkbeck, University of LondonLondonUK
  2. 2.School of MathematicsThe University of New South WalesSydneyAustralia
  3. 3.MathWorksCambridgeUK

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