COMPENSATED CONVEXITY METHODS FOR APPROXIMATIONS AND INTERPOLATIONS OF SAMPLED FUNCTIONS IN EUCLIDEAN SPACES: THEORETICAL FOUNDATIONS

摘要

We introduce Lipschitz continuous and C-1,C-1 geometric approximation and interpolation methods for sampled bounded uniformly continuous functions over compact sets and over complements of bounded open sets in R-n by using compensated convex transforms. Error estimates are provided for the approximations of bounded uniformly continuous functions, of Lipschitz functions, and of C-1,C-1 functions. We also prove that our approximation methods, which are differentiation and integration free and not sensitive to sample type, are stable with respect to the Hausdorff distance between samples.