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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 × aZ 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 → 0, a > 0. For a large class of data functions defined on R x 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 (p - 1) everywhere (in particular, they are smooth on.the hyperplanes {j} x R n , which includes existence of the normal derivatives up to order 2 (p - 1)) 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.

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