In approximation theory, three classical types of results are direct theorems,
Bernstein inequalities, and inverse theorems. In this paper, we include results about
radial basis function (RBF) approximation from all three classes. Bernstein inequalities
are a recent development in the theory of RBF approximation, and on Rd, only
L2 results are known for RBFs with algebraically decaying Fourier transforms (e.g.
the Sobolev splines and thin-plate splines). We will therefore extend what is known
by establishing Lp Bernstein inequalities for RBF networks on Rd. These inequalities
involve bounding a Bessel-potential norm of an RBF network by its corresponding Lp
norm in terms of the separation radius associated with the network. While Bernstein
inequalities have a variety of applications in approximation theory, they are most commonly
used to prove inverse theorems. Therefore, using the Lp Bernstein inequalities
for RBF approximants, we will establish the corresponding inverse theorems. The
direct theorems of this paper relate to approximation in Lp(Rd) by RBFs which are
perturbations of Green's functions. Results of this type are known for certain compact
domains, and results have recently been derived for approximation in Lp(Rd)
by RBFs that are Green's functions. Therefore, we will prove that known results for
approximation in Lp(Rd) hold for a larger class of RBFs. We will then show how this
result can be used to derive rates for approximation by Wendland functions.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-2010-08-8404 |
| Date | 2010 August 1900 |
| Creators | Ward, John P. |
| Contributors | Narcowich, Francis J., Ward, Joseph D. |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | thesis, text |
| Format | application/pdf |
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