Title: 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 other topics. Though existence and
uniqueness of solutions are established for long, we present several 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, (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 allows the description of solutions as global minimizers of an energy
functional.