On Kramer’s Sampling Theorem Associated with General Sturm-Liouville Problems and Lagrange Interpolation

Abstract

Kramer’s sampling theorem (which is a generalization of Shannon’s sampling theorem) and its relationship with Lagrange interpolation is studied. Kramer’s sampling theorem is extended to the case where the interval $( {a,b} )$ is infinite and it is shown that any function that has a sampling expansion in the scope of Kramer’s theorem also has a Lagrange-type interpolation expansion, provided that the kernel associated with Kramer’s theorem arises from a second-order Sturm-Liouville boundary value problem, whether it is regular or singular. This new technique not only reproduces many of the known sampling expansions such as those for the cosine, the finite Hankel, and the continuous Legendre transforms, but also generates new ones such as those for the continuous Jacobi, Laguerre, and Hermite transforms.