Modified Expansion Theorem for Sturm-Liouville problem with transmission conditions

TL;DR

This paper develops an eigenfunction expansion theorem for a discontinuous Sturm-Liouville problem with transmission conditions, including Green's function and uniform convergence results.

Contribution

It introduces a modified expansion theorem specifically for Sturm-Liouville problems with transmission conditions, addressing discontinuities at a point.

Findings

Eigenfunction expansion theorem for the Green's function

Uniform convergence theorem for a class of functions

Extension of classical Sturm-Liouville theory to discontinuous problems

Abstract

This paper is devoted to the derivation of expansion a associated with a discontinuous Sturm-Liouville problems defined on $[-\pi, 0)\cup(0,\pi]$. We derive an eigenfunction expansion theorem for the Green's function of the problem as well as a theorem of uniform convergence of a certain class of functions.