ABSTRACT
We consider interpolation of real-valued functions / defined on a set Ω C JRd, d > 1. These functions axe interpolated on a set X := {χ χ ,. . . , xnx } of Νχ > 1 pairwise distinct points x 1?. . . , xnx in Ω. Interpolation is done by linear combinations of translates Φ(χ — Xj) of a single continuous real-valued function Φ defined on lRd. For various reasons it is sometimes necessary to add the space of d-variate polynomials of order not exceeding m to the interpolating functions. Interpolation is uniquely possible under the assump tion
if p E P ^ satisfies p(x{) = 0 for all x,· E X , then p = 0, (1)
D efin ition 1. A function Φ : JRd —► 1R with Φ(—x) = Φ(χ) is conditionally positive definite of order m on JRd if for all sets X = { x i , . . . , xnx } C lRd with Νχ distinct points and all vectors α := (αχ ,. . . , awx ) € 1RNx with
( 2)
the quadratic form a j a k $ ( x j —Xk) is always nonnegative, and vanishes only if a = 0.
