, Volume 6, Issue 3, pp 225-255

Multivariate cardinal interpolation with radial-basis functions

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Abstract

For a radial-basis functionϕ∶ℛ→ℛ we consider interpolation on an infinite regular lattice
, tof∶ℛ n→ℛ, whereh is the spacing between lattice points and the cardinal function
, satisfiesX(j)=δ oj for allj∈ℒ n. We prove existence and uniqueness of such cardinal functionsX, and we establish polynomial precision properties ofI h 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 ofI hf to sufficiently differentiable functionsf whenh0.
Communicated by Ronald A. DeVore.AMS classification: Primary 41A05, 41A63, 41A25; Secondary 41A30.