Compensated Convexity Methods for Approximations and Interpolations of Sampled Functions in Euclidean Spaces: Theoretical Foundations

TL;DR

This paper presents new compensated convexity-based methods for approximating and interpolating sampled functions in Euclidean spaces, providing theoretical error bounds and stability analysis.

Contribution

It introduces Lipschitz and $C^{1,1}$ approximation techniques using compensated convex transforms with proven error estimates and stability properties.

Findings

Error estimates for various function classes

Methods are differentiation and integration free

Stable with respect to sample Hausdorff distance

Abstract

We introduce Lipschitz continuous and $C^{1,1}$ geometric approximation and interpolation methods for sampled bounded uniformly continuous functions over compact sets and over complements of bounded open sets in $\mathbb{R}^n$ by using compensated convex transforms. Error estimates are provided for the approximations of bounded uniformly continuous functions, of Lipschitz functions, and of $C^{1,1}$ 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.