Scattered Data Interpolation 193 x 1

This paper deals with some basic aspects of scattered data problems. In particular, the following topics are discussed: Self-adjoint scattered data interpolation, sampling sets and interpolation sets, and irregular sampling of shift-invariant spline spaces. Results on band-limited functions are presented as well as results on univariate splines on the real line. x0. Introduction Scattered data interpolation problems are encountered in practical applications when the location of measurements is irregularly (or non-uniformly) distributed. It has been one of the major challenges during the past two decades, in Approximation Theory and in Numerical Mathematics, to analyse these problems, to understand the basic structures involved, and to give solutions in terms of eecient numerical algorithms. Radial basis functions are applied in scattered data interpolation, because they often show surprisingly good approximation properties and because the radial symmetry allows for a quick evaluation of the interpolant. For an account of topics discussed in the radial basis function literature, we may refer to the recent survey paper by Schaback 25] and to the references therein. Our paper does not aim at providing a comparable survey; the reader will recognize that the topics discussed here are more directed into applications in sampling of signals. However, we nd it opportune to include at least some basic notions and ideas of radial basis interpolation and approximation; in this way the diierence to other recently studied problems of irregular sampling will become more transparent. Radial basis interpolation and approximation is often put into its varia-tional formulation in order to see the basic structure of spaces involved. This is in complete analogy to the Ritz-Galerkin approach to Finite Elements, but Surface Fitting and Multiresolution Methods 191 A. ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.