Some general aspects of the sampling theorem

The sampling theorem is recognized as an interpolation formula. Starting from the Lagrange Polynomial, this theorem is developed under conditions which are of broader applicability than those usually stated. Such a view point indicates the essential unity of temporal and frequency domain application. It will also be shown that the theorem is applicable as an exact interpolation formula throughout the complex plane. The basic theorem is extended to include sampling of the first derivative of the function. The concept of band-limited functions is introduced through use of Fourier-Stieltjes representations. This is then shown to be subsumed under the general class of functions which is considered in connection with the interpolation theorems developed. This approach, as presented, readily leads to the establishment of many sampling theorems. It is hoped that this paper will aid the formulation of particularly applicable sampling theorems for specific problems.