Continued fraction solution of Krein's inverse problem

TL;DR

This paper extends the use of continued fractions to solve Krein's inverse problem for strings with more general mass distributions, including an application to diffusion process excursion theory.

Contribution

It adapts Stieltjes' continued fraction method to a broader class of strings beyond purely discrete mass distributions.

Findings

Extended continued fraction method for general strings

Solved inverse problem for new class of strings

Applied method to diffusion process excursions

Abstract

The spectral data of a vibrating string are encoded in its so-called characteristic function. We consider the problem of recovering the distribution of mass along the string from its characteristic function. It is well-known that Stieltjes' continued fraction provides a solution of this inverse problem in the particular case where the distribution of mass is purely discrete. We show how to adapt Stieltjes' method to solve the inverse problem for a related class of strings. An application to the excursion theory of diffusion processes is presented.