Open Problem Concerning Fourier Transforms of Radial Functions in Euclidean Space and on Spheres

  • Jeremy Levesley
  • Simon Hubbert
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 137)

Abstract

Let P n (λ) be the Gegenbauer polynomials, orthogonal on [−1,1] with respect to the weight (1 − t 2)λ−1/2, normalised by
$$P_n^{(\lambda )}(1) = \frac{{\Gamma (n + 2\lambda )}}{{\Gamma (2\lambda )\Gamma (n + 1)}}$$
.

Keywords

Euclidean Space Radial Basis Function Spherical Harmonic Fourier Coefficient Approximation Theory 
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References

  1. [1]
    B. J. C. Baxter, S. Hubbert: Radial basis functions for the sphere. In: Recent Progress in Multivariate Approximation, Internat. Ser. Numer. Math. 137, W. Haußmann, K. Jetter, M. Reimer (eds.), Birkhäuser, Basel 2001, pp. 33–47.CrossRefGoogle Scholar
  2. [2]
    E. W. Cheney, W. A. Light: A Course in Approximation Theory, Pacific Grove, California, 2000.Google Scholar
  3. [3]
    E. M. Stein, G. Weiss: Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N. J., 1990.Google Scholar
  4. [4]
    J. Levesley: Convergence of Euclidean radial basis approximation on spheres, Technical Report 2000/28, Dept. of Mathematics and Computer Science, University of Leicester, Leicester LE1 7RH, 2000.Google Scholar
  5. [5]
    J. Levesley, W. A. Light, D. L. Ragozin, X. Sun: A simple approach to the variational theory for interpolation on spheres. In: New Developments in Approximation Theory, Interat. Ser. Numer. Math. 132, M. W. Müller, M. D. Buhmann, D. H. Mache, M. Felten (eds.), Birkhäuser, Basel 1999, pp. 117–143.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2001

Authors and Affiliations

  • Jeremy Levesley
    • 1
  • Simon Hubbert
    • 2
  1. 1.Department of Mathematics and Computer ScienceUniversity of LeicesterLeicesterEngland
  2. 2.Department of MathematicsImperial CollegeLondonEngland

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