# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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## Polyharmonic splines on grids $\mathbb {Z}\times a\mathbb {Z}^{n}$ and their limitsHTML articles powered by AMS MathViewer

by O. Kounchev and H. Render
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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• 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