Worst-case optimal approximation with increasingly flat Gaussian kernels

Toni Karvonen*, Simo Särkkä

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

1 Citation (Scopus)
66 Downloads (Pure)

Abstract

We study worst-case optimal approximation of positive linear functionals in reproducing kernel Hilbert spaces induced by increasingly flat Gaussian kernels. This provides a new perspective and some generalisations to the problem of interpolation with increasingly flat radial basis functions. When the evaluation points are fixed and unisolvent, we show that the worst-case optimal method converges to a polynomial method. In an additional one-dimensional extension, we allow also the points to be selected optimally and show that in this case convergence is to the unique Gaussian quadrature–type method that achieves the maximal polynomial degree of exactness. The proofs are based on an explicit characterisation of the reproducing kernel Hilbert space of the Gaussian kernel in terms of exponentially damped polynomials.

Original languageEnglish
Article number21
Number of pages17
JournalADVANCES IN COMPUTATIONAL MATHEMATICS
Volume46
Issue number2
DOIs
Publication statusPublished - 6 Mar 2020
MoE publication typeA1 Journal article-refereed

Keywords

  • Gaussian kernel
  • Gaussian quadrature
  • Reproducing kernel Hilbert spaces
  • Worst-case analysis

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