Kramer-type sampling theorems associated with higher-order differential equations

Abstract

For many decades, Kramer’s sampling theorem has been attracting enormous interest in view of its important applications in various branches. In this paper we present a new approach to a Kramer-type theory based on spectral differential equations of higher order on an interval of the real line. Its novelty relies partly on the fact that the corresponding eigenfunctions are orthogonal with respect to a scalar product involving a classical measure together with a point mass at a finite endpoint of the domain. In particular, a new sampling theorem is established, which is associated with a self-adjoint Bessel-type boundary value problem of fourth-order on the interval [0, 1]. Moreover, we consider the Laguerre and Jacobi differential equations and their higher-order generalizations and establish the Green-type formulas of the differential operators as an essential key towards a corresponding sampling theory.