Kramer Sampling Theorem For Multidimensional Signals And Its Relationship With Lagrange-Type Interpolations

Kramer's sampling theorem, which is a generalization of the Whittaker-Shannon-Kotel'nikov (WSK) sampling theorem, enables one to reconstruct functions that are integral transforms of types other than the Fourier one from their sampled values. In this paper, we generalize Kramer's theorem to N dimensions (N greater-than-or-equal-to 1) and show how the kernel function and the sampling points in Kramer's theorem can be generated. We then investigate the relationship between this generalization of Kramer's theorem and N-dimensional versions of both the WSK theorem and the Paley-Wiener interpolation theorem for band-limited signals. It is shown that the sampling series associated with this generalization of Kramer's theorem is nothing more than an N-dimensional Lagrange-type interpolation series.