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Applications of classical approximation theory to periodic basis function networks and computational harmonic analysis

Mhaskar, Hrushikesh N. and Nevai, Paul and Shvarts, Eugene (2013) Applications of classical approximation theory to periodic basis function networks and computational harmonic analysis. Bulletin of Mathematical Sciences, 3 (3). pp. 485-549. ISSN 1664-3615. doi:10.1007/s13373-013-0043-1. https://resolver.caltech.edu/CaltechAUTHORS:20140227-103351879

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Abstract

In this paper, we describe a novel approach to classical approximation theory of periodic univariate and multivariate functions by trigonometric polynomials. While classical wisdom holds that such approximation is too sensitive to the lack of smoothness of the target functions at isolated points, our constructions show how to overcome this problem. We describe applications to approximation by periodic basis function networks, and indicate further research in the direction of Jacobi expansion and approximation on the Euclidean sphere. While the paper is mainly intended to be a survey of our recent research in these directions, several results are proved for the first time here.


Item Type:Article
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URLURL TypeDescription
http://dx.doi.org/10.1007/s13373-013-0043-1DOIArticle
http://link.springer.com/article/10.1007/s13373-013-0043-1PublisherArticle
Additional Information:© 2013 The Author(s). This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. Received: 22 January 2013; Revised: 13 June 2013; Accepted: 17 June 2013; Published online: 9 July 2013. Communicated by S.K. Jain. The research of H.N. Mhaskar was supported, in part, by grant DMS-0908037 from the National Science Foundation and grant W911NF-09-1-0465 from the U.S. Army Research Office. The research of P. Nevai was supported by a KAU grant. We thank the referee for his comments, corrections, critical observations, valuable suggestions, and, in particular, for pointing out [14,32,40,41,72].
Funders:
Funding AgencyGrant Number
NSFDMS-0908037
Army Research Office (ARO)W911NF-09-1-0465
King Abdulaziz UniversityUNSPECIFIED
Issue or Number:3
DOI:10.1007/s13373-013-0043-1
Record Number:CaltechAUTHORS:20140227-103351879
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20140227-103351879
Official Citation:Mhaskar, H., Nevai, P., & Shvarts, E. (2013). Applications of classical approximation theory to periodic basis function networks and computational harmonic analysis. Bulletin of Mathematical Sciences, 3(3), 485-549. doi: 10.1007/s13373-013-0043-1
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:44033
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:28 Feb 2014 00:16
Last Modified:10 Nov 2021 16:46

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