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

Zhang, Kewei; Crooks, Elaine; Orlando, Antonio

Authors

KEWEI ZHANG Kewei.Zhang@nottingham.ac.uk
Professor of Mathematical Analysis

Elaine Crooks

Antonio Orlando



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 http://eprints.nottingham.ac.uk/id/eprint/40889
Publisher URL http://epubs.siam.org/doi/10.1137/15M1045673
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf
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.

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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