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

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.