On sampling theory and eigenvalue problems with an eigenparameter in the boundary conditions

Abstract

This paper is devoted to the investigation of sampling theory associated with second order eigenvalue problems with an eigenparameter appearing in the boundary conditions. We study two cases. The first is when the eigen-parameter appears linearly in all boundary conditions and the second is when it appears only in one condition. We closely follow the analysis derived by C. T. Fulton (1977) to establish the needed relations for the derivations of the sampling theorems including the construction of Green’s function as well as the eigenfunction expansion theorem. We derive sampling representations for transforms whose kernels are either solutions or Green’s functions.