Compensated convexity methods for approximations and interpolations of sampled functions in euclidean spaces: Theoretical foundations

Abstract

We introduce Lipschitz continuous and C1;1 geometric approximation and interpolation methods for sampled bounded uniformly continuous functions over compact sets and over complements of bounded open sets in Rn by using compensated convex transforms. Error estimates are provided for the approximations of bounded uniformly continuous functions, of Lipschitz functions, and of C1;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.