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Polyharmonic splines on grids $\mathbb{Z}\times a\mathbb{Z} ^{n}$ and their limits

Authors: O. Kounchev and H. Render
Journal: Math. Comp. 74 (2005), 1831-1841
MSC (2000): Primary 41A05, 65D10; Secondary 41A15
Published electronically: February 14, 2005
MathSciNet review: 2164099
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Abstract: Radial Basis Functions (RBF) have found a wide area of applications. We consider the case of polyharmonic RBF (called sometimes polyharmonic splines) where the data are on special grids of the form $\mathbb{Z}\times a\mathbb{Z} ^{n}$ having practical importance. The main purpose of the paper is to consider the behavior of the polyharmonic interpolation splines $I_{a}$on such grids for the limiting process $a\rightarrow0,$ $a>0.$ For a large class of data functions defined on $\mathbb{R}\times\mathbb{R} ^{n}$ it turns out that there exists a limit function $I.$ This limit function is shown to be a polyspline of order $p$ on strips. By the theory of polysplines we know that the function $I$ is smooth up to order $2\left( p-1\right) $everywhere (in particular, they are smooth on the hyperplanes $\left\{ j\right\} \times\mathbb{R} ^{n}$, which includes existence of the normal derivatives up to order $2\left( p-1\right))$ while the RBF interpolants $I_{a}$ are smooth only up to the order $2p-n-1.$ The last fact has important consequences for the data smoothing practice.

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Additional Information

O. Kounchev
Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev St. 8, 1113 Sofia, Bulgaria

H. Render
Affiliation: Departamento de Matemáticas y Computatión, Universidad de la Rioja, Edificio Vives, Luis de Ulloa, s/n 26004, Logroño, Spain

Keywords: Radial basis functions, interpolation, polyharmonic splines, polysplines.
Received by editor(s): August 14, 2003
Received by editor(s) in revised form: June 25, 2004
Published electronically: February 14, 2005
Article copyright: © Copyright 2005 American Mathematical Society