Interpolating D ^M-Splines

Abstract

In Chapter 5, we consider the most important example of multivariate splines in the variational spline theory. We mean D ^M-splines. Many results in D ^M-spline theory are due to the works of M.Atteia, J.Duchon, W.Freeden, J.Meigneut, S.L.Sobolev, G.Wahba, etc. A valuable contribution in development of the theory has been made by the authors of this monograph too. It concerns the errors of interpolation for D ^M-splines, their finite-element analogs, D ^M-splines with boundary conditions. The exact rates of convergence were presented in Chapter 4 on the basis of a general technique, but here we make more precise formulations. All error estimates attained in the chapter are given in the Sobolev semi-norms. To get them we prove the so called lemma on the Sobolev functions with condensed zeros. Then this lemma is applied in different situations. The sense of the lemma consists in the following: if a function has a dense set of zeros, and its Sobolev norm or semi-norm is bounded, then this function is very small.