Radial basis functions and corresponding zonal series expansions on the sphere

  • Authors:

  • W. zu Castell;F. Filbir

  • Affiliations:

  • Institute of Biomathematics and Biometry, GSF-National Research Center for Environment and Health, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany;Institute of Biomathematics and Biometry, GSF-National Research Center for Environment and Health, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2005

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Abstract

Since radial positive definite functions on R^d remain positive definite when restricted to the sphere, it is natural to ask for properties of the zonal series expansion of such functions which relate to properties of the Fourier-Bessel transform of the radial function. We show that the decay of the Gegenbauer coefficients is determined by the behavior of the Fourier-Bessel transform at the origin.