Finite Blaschke products with prescribed critical points, Stieltjes polynomials, and moment problems

Abstract

The determination of a finite Blaschke product from its critical points is a well-known problem with interrelations to several other topics. Though existence and uniqueness of solutions are established for long, we present new aspects which have not yet been explored to their full extent. In particular, we show that the following three problems are equivalent: (i) determining a finite Blaschke product from its critical points, (ii) finding the equilibrium position of moveable point charges interacting with a special configuration of fixed charges, and (iii) solving a moment problem for the canonical representation of power moments on the real axis. These equivalences are not only of theoretical interest, but also open up new perspectives for the design of algorithms. For instance, the second problem is closely linked to the determination of certain Stieltjes and Van Vleck polynomials for a second order ODE and characterizes solutions as global minimizers of an energy functional.