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

Alatawi, Maryam; Zhang, Kewei

Abstract

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.

Citation

Alatawi, M., & Zhang, K. (2020). Hausdorff Stability and Error Estimates for Compensated Convexity Based Methods for Approximation and Interpolation for Functions in Rn. Journal of Convex Analysis, 27(4),

• Journal Article Type: Article
• Acceptance Date: Feb 9, 2020
• Publication Date: 2020
• Deposit Date: May 5, 2020
• Journal: Journal of Convex Analysis
• Print ISSN: 0944-6532
• Electronic ISSN: 2363-6394
• Publisher: Heldermann Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 27
• Issue: 4
• Keywords: Compensated convex transforms, compact samples, complement of bounded open sets interpolation, approximation, inpainting, bounded functions, Cα-functions, C1,β-functions, error estimates, Hausdorff stability, Hausdorff distance, convex density radius.
• Public URL: https://nottingham-repository.worktribe.com/output/4202362
• Publisher URL: http://www.heldermann.de/JCA/JCA27/JCA274/jca27069.htm
• Related Public URLs: http://www.heldermann.de/JCA/jcacover.htm
• Additional Information: Joint work with Maryam Alatawi