A New Role of Green′s Function in Interpolation and Sampling Theory

Abstract

One of the most fundamental theorems in interpolation and sampling theory is the Whittaker-Shannon-Kotel′nikov sampling theorem, which gives an algorithm for reconstructing functions that are Fourier transforms of functions with compact supports, from their sampled values at a discrete set of points. Functions that are integral transforms of types other than the Fourier one, may also be reconstructed from their sampled values at a discrete set of points by using a generalization of the Whittaker-Shannon-Kotel′nikov sampling theorem, known as Kramer′s sampling theorem. The main assumption in Kramer′s sampling theorem is that the kernel of the integral transform to be reconstructed has to arise from a self-adjoint boundary-value problem whose eigenvalues are all simple and the eigenfunctions are all generated by one single function. In this paper, we derive a Kramer-type sampling theorem for a larger class of integral transforms than that considered by Kramer′s theorem, but under less stringent assumptions. For example, the kernel of any of these integral transforms to be reconstructed from its sampled values may arise from a non-self-adjoint boundary-value problem with repeated eigenvalues and eigenfunctions that are not necessarily generated by one single function. We only require that the eigenvalues be simple poles of the associated Green′s function. Our technique is based on the use of Green′s functions of non-self-adjoint boundary-value problems to reconstruct the interpolating functions.