Polyharmonic splines on grids ℤ x aℤn and their limits

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 Z× aZn having practical importance. The main purpose of the paper is to consider the behavior of the polyharmonic interpolation splines Ia on such grids for the limiting process a → 0, a > 0. For a large class of data functions defined on R× Rn 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 (p− 1) everywhere (in particular, they are smooth on the hyperplanes {j} × Rn, which includes existence of the normal derivatives up to order 2 (p− 1)) while the RBF interpolants Ia are smooth only up to the order 2p− n− 1. The last fact has important consequences for the data smoothing practice.