Lower Bounds for Norms of Inverses of Interpolation Matrices for Radial Basis Functions

Abstract

Interpolation of scattered data at distinct points xI,..., xn ∈ Rd by linear combinations of translates Φ(||x − xj||2) of a radial basis function Φ : R≥ 0 → R requires the solution of a linear system with the n by n distance matrix A ≔ (Φ(||xi − xj||2). Recent results of Ball, Narcowich and Ward, using Laplace transform methods, provide upper bounds for ||A−1||2, while Ball, Sivakumar, and Ward constructed examples with regularly spaced points to get special lower bounds. This paper proves general lower bounds by application of results of classical approximation theory. The bounds increase with the smoothness of Φ. In most cases, they leave no more than a factor of n−2 to be gained by optimization of data placement, starting from regularly distributed data. This follows from comparison with results of Ball, Baxter, Sivakumar, and Ward for points on scaled integer lattices and supports the hypothesis that regularly spaced data are near-optimal, as far as the condition of the matrix A is concerned.