Sampling Theorems for Sturm Liouville Problem with Moving Discontinuity Points

TL;DR

This paper develops sampling theorems for a Sturm-Liouville problem with moving symmetric discontinuities, transmission conditions, and an eigenparameter in the boundary, expanding the theoretical framework for such differential equations.

Contribution

It introduces a novel Sturm-Liouville problem with dynamic discontinuities and derives sampling theorems using Green's functions and solution kernels.

Findings

Established relations for sampling theorems

Constructed Green's function for the problem

Derived sampling representations for transforms

Abstract

In this paper, we investigate the sampling analysis for a new Sturm-Liouville problem with symmetrically located discontinuities which are defined to depending on a neighborhood of a midpoint of the interval. Also the problem has transmission conditions at these points of discontinuity and includes an eigenparameter in a boundary condition. We establish briefly the needed relations for the derivations of the sampling theorems and construct Green's function for the problem. Then we derive sampling representations for transforms whose kernels are either solutions or Green's functions.