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An inverse theorem for compact Lipschitz regions in $ \mathbb{R}^d$ using localized kernel bases


Authors: T. Hangelbroek, F. J. Narcowich, C. Rieger and J. D. Ward
Journal: Math. Comp.
MSC (2010): Primary 41A17, 41A27, 41A63
DOI: https://doi.org/10.1090/mcom/3256
Published electronically: October 17, 2017
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Abstract: While inverse estimates in the context of radial basis function approximation on boundary-free domains have been known for at least ten years, such theorems for the more important and difficult setting of bounded domains have been notably absent. This article develops inverse estimates for finite dimensional spaces arising in radial basis function approximation and meshless methods. The inverse estimates we consider control Sobolev norms of linear combinations of a localized basis by the $ L_p$ norm over a bounded domain. The localized basis is generated by forming local Lagrange functions for certain types of RBFs (namely Matérn and surface spline RBFs). In this way it extends the boundary-free construction recently presented by Fuselier, Hangelbroek and Narcowich [Localized bases for kernel spaces on the unit sphere, SIAM J. Numer. Anal. 51 (2013), no. 5, 2358-2562].


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Additional Information

T. Hangelbroek
Affiliation: Department of Mathematics, University of Hawai\kern.05em‘\kern.05emi – \Manoa, 2565 McCarthy Mall, Honolulu, Hawaii
Email: hangelbr@math.hawaii.edu

F. J. Narcowich
Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843
Email: fnarc@math.tamu.edu

C. Rieger
Affiliation: Institut für Numerische Simulation, Universität Bonn, Wegelerstrasse 6, 53115 Bonn, Germany
Email: rieger@ins.uni-bonn.de

J. D. Ward
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: jward@math.tamu.edu

DOI: https://doi.org/10.1090/mcom/3256
Keywords: Radial basis functions, Sobolov spaces, Bernstein and Nikolskii inequalities, trace estimate.
Received by editor(s): August 25, 2015
Received by editor(s) in revised form: May 19, 2016, October 14, 2016, and December 28, 2016
Published electronically: October 17, 2017
Additional Notes: The first author’s research was supported by grant DMS-1413726 from the National Science Foundation.
The second author’s research was supported by grant DMS-1514789 from the National Science Foundation
The third author’s research was supported by Collaborative Research Centre (SFB) 1060: The Mathematics of Emergent Effects, of the Deutsche Forschungsgemeinschaft
The fourth author’s research was supported by grant DMS-1514789 from the National Science Foundation.
Article copyright: © Copyright 2017 American Mathematical Society