Stieltjes Functions and Hurwitz Stable Entire Functions

TL;DR

This paper extends the concept of stability from polynomials to entire functions, introducing Hurwitz stability and providing methods to construct such functions from Stieltjes and Laguerre-Pólya class functions.

Contribution

It introduces Hurwitz stability for entire functions and presents two new theorems for constructing Hurwitz stable functions from existing classes.

Findings

Established a method to construct Hurwitz stable functions from Stieltjes class functions.

Presented a second approach using Laguerre-Pólya class functions.

Extended stability concepts from polynomials to entire functions.

Abstract

The concept of stability, originally introduced for polynomials, will be extended to apply to the class of entire functions. This generalization will be called Hurwitz stablility and the class of Hurwitz stable functions will serve as the main focus of this paper. A first theorem will show how, given a function of either of the Stieltjes classes, a Hurwitz stable function might be constructed. A second approach to constructing Hurwitz stable functions, based on using additional functions from the Laguerre-P\'{o}lya class, will be presented in a second theorem.