Discontinuous Sturm‐Liouville Problems and Associated Sampling Theories

Yuming Shi,M M Tharwat

doi:10.1155/2011/610232

Abstract

This paper investigates the sampling analysis associated with discontinuous Sturm‐Liouville problems with eigenvalue parameters in two boundary conditions and with transmission conditions at the point of discontinuity. We closely follow the analysis derived by 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. In the special case, when our problem is continuous, the obtained results coincide with the corresponding results in the work of Annaby and Tharwat (2006).

Highlights

The recovery of entire functions from a discrete sequence of points is an important problem from mathematical and practical points of view
It is shown that Kramer’s expansion 1.9 could be written as a Lagrange-type interpolation formula if K ·, t and tk are extracted from ordinary differential operators, see the survey 18 and the references cited therein
We prove that integral transforms associated with secondorder eigenvalue problems with an eigenparameter appearing in the boundary conditions and with an internal point of discontinuity can be reconstructed in a sampling form of Lagrange interpolation type

Summary

Introduction

The recovery of entire functions from a discrete sequence of points is an important problem from mathematical and practical points of view. Starting from a function theory approach, cf , it is proved in that if K x, t , x ∈ I, t ∈ C satisfies some analyticity conditions, Kramer’s sampling formula 1.9 turns out to be a Lagrange interpolation one, see 15–17. In another direction, it is shown that Kramer’s expansion 1.9 could be written as a Lagrange-type interpolation formula if K ·, t and tk are extracted from ordinary differential operators, see the survey 18 and the references cited therein. We prove that integral transforms associated with secondorder eigenvalue problems with an eigenparameter appearing in the boundary conditions and with an internal point of discontinuity can be reconstructed in a sampling form of Lagrange interpolation type. We derive two sampling theorems using solutions and Green’s function, respectively

The Eigenvalue Problem

Asymptotic Formulas of Eigenvalues and Eigenfunctions

The Sampling Theorem

28 The Green’s function has the following form