Convergence of increasingly flat radial basis interpolants to polynomial interpolants

Yeon Ju Lee, Gang Joon Yoon, Jungho Yoon

Research output: Contribution to journalArticlepeer-review

26 Scopus citations

Abstract

In this paper, we study the convergence behavior of interpolants by smooth radial basis functions to polynomial interpolants in ℝd as the radial basis functions are scaled to be increasingly flat. Larsson and Fornberg [Comput. Math. Appl., 49 (2005), pp. 103-130] conjectured a sufficient property for this convergence, and they also conjectured that Bessel radial functions do not satisfy this property. First, in the case of positive definite radial functions, we prove both conjectures by Larsson and Fornberg for the convergence of increasingly flat radial function interpolants. Next, we extend the results to the case of conditionally positive definite radial functions of order m > 0.

Original languageEnglish
Pages (from-to)537-553
Number of pages17
JournalSIAM Journal on Mathematical Analysis
Volume39
Issue number2
DOIs
StatePublished - 2007

Keywords

  • Conditionally positive definite function
  • Interpolation
  • Polynomial
  • Radial basis function

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