Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting

Authors:
Francis J. Narcowich, Joseph D. Ward and Holger Wendland

Journal:
Math. Comp. **74** (2005), 743-763

MSC (2000):
Primary 41A25; Secondary 41A05, 41A63

DOI:
https://doi.org/10.1090/S0025-5718-04-01708-9

Published electronically:
August 20, 2004

MathSciNet review:
2114646

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we discuss Sobolev bounds on functions that vanish at scattered points in a bounded, Lipschitz domain that satisfies a uniform interior cone condition. The Sobolev spaces involved may have fractional as well as integer order. We then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions (RBFs). These estimates include situations in which the target function does *not* belong to the native space of the RBF.

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Additional Information

**Francis J. Narcowich**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Email:
fnarc@math.tamu.edu

**Joseph D. Ward**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Email:
jward@math.tamu.edu

**Holger Wendland**

Affiliation:
Universität Göttingen, Lotzestrasse 16-18, D-37083, Göttingen, Germany

Email:
wendland@math.uni-goettingen.de

DOI:
https://doi.org/10.1090/S0025-5718-04-01708-9

Keywords:
Radial basis functions,
Sobolev error estimates,
scattered zeros,
scattered data.

Received by editor(s):
July 21, 2003

Published electronically:
August 20, 2004

Additional Notes:
Research of the first author was supported by grant DMS-0204449 from the National Science Foundation.

Research of the second author was supported by grants DMS-9971276 and DMS-0204449 from the National Science Foundation.

Article copyright:
© Copyright 2004
American Mathematical Society