ON THE SENSITIVITY OF RADIAL BASIS INTERPOLATION TO MINIMAL DATA SEPARATION DISTANCE
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abstract
Motivated by the problem of multivariate scattered data interpolation, much interest has centered on interpolation by functions of the form {Mathematical expression} where g:R + R is some prescribed function. For a wide range of functions g, it is known that the interpolation matrices A=g(x i -x j ) i,j=1N are invertible for given distinct data points x 1 , x 2 ,..., x N . More recently, progress has been made in quantifying these interpolation methods, in the sense of estimating the (l 2 ) norms of the inverses of these interpolation matrices as well as their condition numbers. In particular, given a suitable function g:R + R, and data in R s having minimal separation q, there exists a function h s :R + R + , which depends only on g and s, and a constant C s , which depends only on s, such that the inverse of the associated interpolation matrix A satisfies the estimate {norm of matrix}A -1 {norm of matrix}C s h s (q). The present paper seeks "converse" results to the inequality given above. That is, given a suitable function g, a spatial dimension s, and a parameter q>0 (which is usually assumed to be small), it is shown that there exists a data set in R s having minimal separation q, a constant {Mathematical expression} depending only on s, and a function k s (q), such that the inverse of the interpolation matrix A associated with this data set satisfies {Mathematical expression}. In some cases, it is seen that h s (q)=k s (q), so the bounds are optimal up to constants. In certain others, k s (q) is less than h s (q), but nevertheless exhibits a behavior comparable to that of h s (q). That is, even in these cases, the bounds are close to being optimal. 1992 Springer-Verlag New York Inc.
