Sampling theory approach to prolate spheroidal wavefunctions

We use the Whittaker-Shannon sampling theorem to show that the eigenvalue problem for the sinc-kernel is equivalent to a discrete eigenvalue problem. The well-known eigenfunctions, namely, the prolate spheroidal wavefunctions, their corresponding eigenvalues and the orthogonality and completeness properties are determined without invoking the prolate spheroidal differential equation. This analysis based on the sampling theorem may be used for calculating the eigenvalues and eigenfunctions of bandlimited kernels in general as we illustrate with an additional example of the sinc(2)-kernel.