On the Accuracy of Surface Spline Approximation and Interpolation to Bump Functions

Let be the closure of a bounded open set in R d and, for a suu-ciently large integer , let f 2 C (() be a real-valued \bump" function, i.e. supp(f) int((). First, for each h > 0, we construct a surface spline function h whose centres are the vertices of the grid V h = \ hZ d , such that h approximates f uniformly over with the maximal asymp-totic accuracy rate for h ! 0. Second, if`1; `2; : : : ; `n are the Lagrange functions for surface spline interpolation on the grid V h , we prove that maxx2 P n j=1`2 j (x) is bounded above independently of the mesh-size h. An interesting consequence of these two results for the case of interpolation on V h to the values of a bump data function f is obtained by means of the Lebesgue inequality.