Sampling theorems associated with fourth- and higher-order self-adjoint eigenvalue problems

Abstract

Consider a regular self-adjoint eigenvalue problem of nth order, and the associated (Whittaker–Shannon)–Kramer sampling series, the samples being taken at the eigenvalues, and the sinc-function being replaced by the generalized Fourier coefficients of the kernel (arising from the eigenvalue problem) with respect to the complete set of (orthogonal) eigenfunctions. The major aim of this paper is to show that each such sampling expansion can be written as a Lagrange interpolation series, provided the kernel satisfies suitable conditions. Two concrete fourth-order eigenvalue problems are examined, a regular one as an application of the general result, and a singular one. The problem whether the two resulting sampling series are really new (or can be reduced to already known results) is in part open.