Constructive Approximation

, Volume 6, Issue 3, pp 225–255

Multivariate cardinal interpolation with radial-basis functions

  • M. D. Buhmann

DOI: 10.1007/BF01890410

Cite this article as:
Buhmann, M.D. Constr. Approx (1990) 6: 225. doi:10.1007/BF01890410


For a radial-basis functionϕ∶ℛ→ℛ we consider interpolation on an infinite regular lattice
, tof∶ℛn→ℛ, whereh is the spacing between lattice points and the cardinal function
, satisfiesX(j)=δoj for allj∈ℒn. We prove existence and uniqueness of such cardinal functionsX, and we establish polynomial precision properties ofIh for a class of radial-basis functions which includes\(\varphi (r) = r^{2q + 1} \),\(\varphi (r) = r^{2q} \log r,\varphi (r) = \sqrt {r^2 + c^2 } \), and\(\varphi (r) = 1/\sqrt {r^2 + c^2 } \) whereq∈ℒ+. We also deduce convergence orders ofIhf to sufficiently differentiable functionsf whenh0.

Key words and phrases

Multivariate interpolationMultivariate approximationRadial-basis functionsOrders of convergence

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

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