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

• Published in: Mathematics of computation 2005-10, Vol.74 (252), p.1831-1842
• Main Authors: Kounchev, O., Render, H.
• Format: Article
• Language: English
• ISSN: 0025-5718; 1088-6842
• DOI: 10.1090/S0025-5718-05-01753-9

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 \emph{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.