Cardinal interpolation with periodic polysplines on strips

Abstract

We obtain a multivariate extension of a classical result of Schoenberg on cardinal spline interpolation. Specifically, we prove the existence of a unique function in $C^{2p-2}\left( \mathbb{R}^{n+1}\right) $ , polyharmonic of order p on each strip $\left( j,j+1\right) \times\mathbb{R}^{n}$ , $j\in\mathbb{Z}$ , and periodic in its last n variables, whose restriction to the parallel hyperplanes $\left\{ j\right\} \times\mathbb{R}^{n}$ , $j\in\mathbb{Z}$ , coincides with a prescribed sequence of n-variate periodic data functions satisfying a growth condition in $\left\vert j\right\vert $ . The constructive proof is based on separation of variables and on Micchelli’s theory of univariate cardinal $\mathcal{L}$ -splines. Keywords: cardinal $\mathcal{L}$ -splines, polyharmonic functions, multivariable interpolation Mathematics Subject Classification (2000): 41A05, 41A15, 41A63