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Journal of Convex Analysis 27 (2020), No. 4, 1277--1302
Copyright Heldermann Verlag 2020

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

Maryam Alatawi
Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia

Kewei Zhang
School of Math. Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom

We establish error estimates and Hausdorff stability for approximations and interpolations for sampled functions in Rn 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 Cα or C1,β. 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.

Keywords: Compensated convex transforms, compact samples, complement of bounded open sets, interpolation, approximation, inpainting, bounded functions, error estimates, Hausdorff stability, Hausdorff distance, convex density radius.

MSC: 65K10, 26B25 52A41, 94A20, 94A08.

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