Cardinal interpolation with periodic polysplines on strips

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 C2p-2 (Rn+1), polyharmonic of order p on each strip (j, j + 1) x R-n, j is an element of Z, and periodic in its last n variables, whose restriction to the parallel hyperplanes {j} x R-n, j is an element of Z, coincides with a prescribed sequence of n-variate periodic data functions satisfying a growth condition in vertical bar j vertical bar . The constructive proof is based on separation of Variables and on Micchelli's theory of univariate cardinal L-splines.