Constructive Approximation

, Volume 6, Issue 3, pp 225–255

# Multivariate cardinal interpolation with radial-basis functions

• M. D. Buhmann
Article

## Abstract

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 interpolation Multivariate approximation Radial-basis functions Orders of convergence

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