Stieltjes Functions and Hurwitz Stable Functions

Specialeforsvar ved Jacob Stevne Jørgensen

Titel: Stieltjes Functions and Hurwitz Stable Functions

Abstract:  This thesis studies the relation between functions of the Stieltjes classes and Hurwitz stable functions. We start by introducing the classes of Pick and Stieltjes functions and their basic properties. Afterwards we introduce entire functions with main focus on indicators and the closely related topics of ρ-trigonometrically convex functions and supporting functions of compact, convex sets of the plane. Then we define the notion of Hurwitz stability, starting with polynomials and generalising to entire functions, and thus we are ready to formulate and prove our first main result: To construct a Hurwitz stable entire function, given a function of one of the Stieltjes classes. The last part of the thesis aims at generalising the first main theorem. We start by presenting results on the Laguerre-Pólya class, the Cartwright class C, a theorem of Hayman, Levin’s class P, and a composition theorem for polynomials. All this theory enable us to prove our second main result: To construct a function of class P, given a function of one of the Stieltjes classes and a function of type I in the Laguerre-Pólya class

Vejleder: Henrik Laurberg Pedersen
Censor:   Søren Fournais, Aarhus Universitet