A problem of L. L. Campbell on the equivalence of the Kramer and Shannon sampling theorems

Abstract

A result concerning the equivalence of the sampling theorems of Kramer and Whittaker-Shannon-Kotelnikov is presented in case the associated kernels are Bessel functions or Jacobi functions. The general problem is open. A generalized Mehler-Dirichlet formula for the Jacobi functions is used to give the proof in this case. For the Bessel functions, a generalization of Poisson's integral is needed. It turns out that in both cases, Kramer's theorem gives nothing more than Shannon's theorem in the sense that each function that can be sampled by Kramer's theorem can also be reconstructed by the classical Shannon sampling result.