K-order Sampling ofN-dimensional Band-limited Functions†

Abstract

The work of II previous paper, utilizing the gradient in sampling theory, is generalized further to include the sampling of a function and its partial derivatives up to order K ≥ 1. The reconstruction of the sampled function f(t) has the form of a. sum of truncated Taylor series expansions about each of the sample points which lie on a periodic lattice. The single function g(t), of the vector from the sample point to a generic one, which multiplies each series, hag a J+1ourier transform G(r) which must satisfy sets of partial differential equations on the ’ lattice sets ’ of the support of F(r). The necessary and sufficient conditions which insure the validity of the reconstruction demand that the ‘ associated set ’ of each lattice set be among the zeros of n certain kind of polynomial function which in turn can act as u solution for G(r) on the lattice set.