On the sensitivity of radial basis interpolation to minimal data separation distance

Abstract

Motivated by the problem of multivariate scattered data interpolation, much interest has centered on interpolation by functions of the form $$f(x) = \sum\limits_{j = 1}^N {a_j g(\parallel x - x_j \parallel ),x \in R^s }$$ whereg:R + →R is some prescribed function. For a wide range of functionsg, it is known that the interpolation matricesA=g(∥x i −x j ∥) i,j=1 N are invertible for given distinct data pointsx 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 functiong:R + →R, and data inR s having minimal separationq, there exists a functionh s:R + →R +, which depends only ong ands, and a constantC s , which depends only ons, such that the inverse of the associated interpolation matrixA satisfies the estimate ‖A −1‖≤C s h s (q). The present paper seeks “converse” results to the inequality given above. That is, given a suitable functiong, a spatial dimensions, and a parameterq>0 (which is usually assumed to be small), it is shown that there exists a data set inR s having minimal separationq, a constant\(\tilde C_s\) depending only ons, and a functionk s (q), such that the inverse of the interpolation matrixA associated with this data set satisfies\(\parallel A^{ - 1} \parallel \geqslant \tilde C_s k_s (q)\). In some cases, it is seen thath s(q)=k s (q), so the bounds are optimal up to constants. In certain others,k s (q) is less thanh s (q), but nevertheless exhibits a behavior comparable to that ofh s (q). That is, even in these cases, the bounds are close to being optimal.