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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 $\left( j,j+1\right) \times\mathbb{R}^{n}$ , , and periodic in its last n variables, whose restriction to the parallel hyperplanes...