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We introduce Lipschitz continuous and $C^{1,1}$ geometric approximation andinterpolation methods for sampled bounded uniformly continuous functions overcompact sets and over complements of bounded open sets in $\mathbb{R}^n$ byusing compensated convex transforms. Error estimates are provided for theapproximations of bounded uniformly continuous functions, of Lipschitzfunctions, and of $C^{1,1}$ functions. We also prove that our approximationmethods, which are differentiation and integration free and not sensitive tosample type, are stable with respect to the Hausdorff distance between samples.

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