Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

$ L^p$ Bernstein estimates and approximation by spherical basis functions


Authors: H. N. Mhaskar, F. J. Narcowich, J. Prestin and J. D. Ward
Journal: Math. Comp. 79 (2010), 1647-1679
MSC (2000): Primary 41A17, 41A27, 41A63, 42C15
DOI: https://doi.org/10.1090/S0025-5718-09-02322-9
Published electronically: December 2, 2009
MathSciNet review: 2630006
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to establish $ L^p$ error estimates, a Bernstein inequality, and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit $ n$-sphere. In particular, the Bernstein inequality estimates $ L^p$ Bessel-potential Sobolev norms of functions in this space in terms of the minimal separation and the $ L^p$ norm of the function itself. An important step in its proof involves measuring the $ L^p$ stability of functions in the approximating space in terms of the $ \ell^p$ norm of the coefficients involved. As an application of the Bernstein inequality, we derive inverse theorems for SBF approximation in the $ L^P$ norm. Finally, we give a new characterization of Besov spaces on the $ n$-sphere in terms of spaces of SBFs.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 41A17, 41A27, 41A63, 42C15

Retrieve articles in all journals with MSC (2000): 41A17, 41A27, 41A63, 42C15


Additional Information

H. N. Mhaskar
Affiliation: Department of Mathematics, California State University, Los Angeles, California 90032

F. J. Narcowich
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

J. Prestin
Affiliation: Institute of Mathematics, University of Lübeck, Wallstrasse 40, 23560, Lübeck, Germany

J. D. Ward
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

DOI: https://doi.org/10.1090/S0025-5718-09-02322-9
Keywords: Sphere, Bernstein estimates, approximation, spherical basis functions.
Received by editor(s): October 15, 2008
Received by editor(s) in revised form: July 8, 2009
Published electronically: December 2, 2009
Additional Notes: The research of the first author was supported by grant DMS-0605209 from the National Science Foundation and grant W911NF-04-1-0339 from the U.S. Army Research Office.
The research of the second author was supported by grants DMS-0504353 and DMS-0807033 from the National Science Foundation.
The research of the fourth was supported by grants DMS-0504353 and DMS-0807033 from the National Science Foundation.
Article copyright: © Copyright 2009 American Mathematical Society