Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: theoretical foundations

Zhang, Kewei; Crooks, Elaine; Orlando, Antonio

Abstract

We introduce Lipschitz continuous and C¹,¹ geometric approximation and interpolation methods for sampled bounded uniformly continuous functions over compact sets and over complements of bounded open sets in Rn by using compensated convex transforms. Error estimates are provided for the approximations of bounded uniformly continuous functions, of Lipschitz functions, and of C1,1 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.

Citation

Zhang, K., Crooks, E., & Orlando, A. (in press). Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: theoretical foundations. SIAM Journal on Mathematical Analysis, 48(6), https://doi.org/10.1137/15M1045673

• Journal Article Type: Article
• Acceptance Date: Sep 27, 2016
• Online Publication Date: Dec 8, 2016
• Deposit Date: Feb 28, 2017
• Publicly Available Date: Feb 28, 2017
• Journal: SIAM Journal on Mathematical Analysis
• Print ISSN: 0036-1410
• Electronic ISSN: 1095-7154
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 48
• Issue: 6
• DOI: https://doi.org/10.1137/15M1045673
• Keywords: interpolation, approximation, compensated convex transforms, Lipschitz functions, local-Lipschitz approximation, Hausdorff stability, error estimates
• Public URL: https://nottingham-repository.worktribe.com/output/836204
• Publisher URL: http://epubs.siam.org/doi/10.1137/15M1045673
• Additional Information: First published in SIAM Journal on Mathematical Analysis in vol. 48, no. 6, published by the Society for Industrial and Applied Mathematics. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
• Contract Date: Feb 28, 2017