Cardinal Interpolation with Radial Basis Functions: An Integral Transform Approach

  • M. D. Buhmann
Part of the International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique book series (ISNM, volume 90)


In this paper we use asymptotic expansions of certain integral transforms in order to derive conditions on a radial basis function ø: R≥0→R that imply the existence of a cardinal function
$$x\left( x \right)=\sum\limits_{k\in {{Z}^{n}}}{{{c}_{k}}\phi \left( \parallel x-k\parallel \right)},x\in {{R}^{n}},$$
which satisfies x(l) = δol for all l ∈ Zn. We also study the rate of decay of |X(x)| for large ‖x‖ and the polynomial recovery of interpolation on Zn using this cardinal function. The conditions hold for many important examples of radial basis functions, such as the multiquadrics and related radial functions, and in contrast to some earlier work by the author they are expressed in terms of asymptotic properties of ø rather than in terms of its Fourier transform.


Asymptotic Expansion Radial Basis Function Meromorphic Function Nonnegative Integer Simple Polis 
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Copyright information

© Birkhäuser Verlag Basel 1989

Authors and Affiliations

  • M. D. Buhmann
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeEngland

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