The Gradient in the Sampling of N-dimensional Band-limited Functions†

Abstract

Two previous generalizations of the Whittaker-Shannon sampling theorem are combined in a -way which emphasizes the value of the gradient of a function defined on a space of N dimensions. The band-limited functions are sampled, with their gradient, on a sampling lattice which results from an appropriate choice of the corresponding reciprocal lattice. Necessary and sufficient conditions, for a particular synthesis of the functions, are given which govern the relationship between the reciprocal lattice and the bounded support of the Fourier transforms of the functions in question. These conditions are satisfied by the corresponding conditions of the N-dimensional theory when the gradient is not sampled. When the gradient is sampled the conditions are loss stringent and thus allow for an additional economy in the sampling. Two two-dimensional examples are worked out in detail to illustrate the results of the theory.