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

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- by O. Kounchev and H. Render PDF
- Math. Comp.
**74**(2005), 1831-1841 Request permission

## 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 $a\rightarrow 0,$ $a>0.$ 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
- Email: kounchev@math.bas.bg; kounchev@math.uni-duisburg.de
**H. Render**- Affiliation: Departamento de Matemáticas y Computatión, Universidad de la Rioja, Edificio Vives, Luis de Ulloa, s/n 26004, Logroño, Spain
- MR Author ID: 268351
- Email: render@math.uni-duisburg.de; herender@dmc.unirioja.es
- Received by editor(s): August 14, 2003
- Received by editor(s) in revised form: June 25, 2004
- Published electronically: February 14, 2005
- © Copyright 2005 American Mathematical Society
- Journal: Math. Comp.
**74**(2005), 1831-1841 - MSC (2000): Primary 41A05, 65D10; Secondary 41A15
- DOI: https://doi.org/10.1090/S0025-5718-05-01753-9
- MathSciNet review: 2164099