Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 41A05, 65D10, 41A15

Retrieve articles in all journals with MSC (2000): 41A05, 65D10, 41A15

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
MR Author ID: 268351

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