In Chapters 8 and 9 we shall extend some of the results obtained in Chapters 4 through 6 to higher dimensions. We begin this chapter by generalizing Kramer’s sampling theorem to N(N ≥ 1) dimensions and then demonstrate, as in the one-dimensional case, how the kernel function K(x, λ) and the sampling points { λ n } n = 0 ∞ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315136905/42045113-3bdf-4714-93f0-8e392d85701f/content/eq1186.tif"/> arise naturally when we solve certain Dirichletor Neumann-type boundary-value problems. We then investigate the relationship between this generalization of Kramer’s theorem on the one hand and N-dimensional versions of both the Whittaker-Shannon-Kotel’nikov (WSK) sampling theorem (Theorem 3.6) and the Paley-Wiener interpolation theorem for band-limited functions (Theorems 3.7 and 3.8) on the other. It will be shown that the sampling series associated with this generalization of Kramer’s theorem is again nothing more than a Lagrange-type interpolation series.