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Numerical Algorithms

, Volume 70, Issue 4, pp 709–726 | Cite as

On spectral distribution of kernel matrices related to radial basis functions

  • Andrew J. Wathen
  • Shengxin Zhu
Original Paper

Abstract

This paper focuses on spectral distribution of kernel matrices related to radial basis functions. By relating a contemporary finite-dimensional linear algebra problem to a classical problem on infinite-dimensional linear integral operator, the paper shows how the spectral distribution of a kernel matrix relates to the smoothness of the underlying kernel function. The asymptotic behaviour of the eigenvalues of a infinite-dimensional kernel operator are studied from a perspective of low rank approximation—approximating an integral operator in terms of Fourier series or Chebyshev series truncations. Further, we study how the spectral distribution of interpolation matrices of an infinite smooth kernel with flat limit depends on the geometric property of the underlying interpolation points. In particularly, the paper discusses the analytic eigenvalue distribution of Gaussian kernels, which has important application on stably computing of Gaussian radial basis functions.

Keywords

Eigenvalues Radial basis functions Spectral distribution Integral equation of the first kind 

Mathematics Subject Classification (2010)

42A10 45A25 45B05 45C05 47A52 47A75 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Numerical Analysis GroupThe University of OxfordOxfordEngland
  2. 2.Laboratory of Computational PhysicsInstitute of Applied Physics and Computational MathematicsBeijingChina

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