Hausdorff Stability and Error Estimates for Compensated Convexity Based Methods for Approximation and Interpolation for Functions in ℝⁿ

We establish error estimates and Hausdorff stability for approximations and interpolations for sampled functions in $\mathbb{R}^n$ by using compensated convex transforms introduced previously by K. Zhang [Compensated convexity and its applications, Ann. l'Institut H. Poincaré (C), Non Linear Analysis 25(4) (2008) 743–771]. We generalize the sharp error estimates obtained recently by K. Zhang, E. Crooks, and A. Orlando [Compensated convexity methods for approximations and interpolations of sampled functions in euclidean spaces: Theoretical foundations, SIAM J. Math. Analysis 48(6) (2016) 4126–4154] to cases when the underlying functions are in the class of $C^\alpha$ or $C^{1,\beta}$. We also establish Hausdorff stability when the functions involved are just assumed to be bounded. The stability of our approximations in this paper is with respect to the Hausdorff distance between graphs of the sampled functions.