Mitja Nedic: The convex combination problem for Herglotz-Nevanlinna functions in two variables

Abstract: Herglotz-Nevanlinna functions are holomorphic functions defined in the poly-upper half-plane having non-negative imaginary part. Any such function admits an integral representation formula involving a real number, a vector of non-negative numbers and a positive Borel measure satisfying certain properties. These parameters are of great interest as they completely characterize this class of functions.

The convex combination problem for Herglotz-Nevanlinna functions asks whether we can relate the parameters of a Herglotz-Nevanlinna function in one variable to the parameters of a Herglotz-Nevanlinna function in several variables, if the several-variable function was built out of the one-variable function by replacing the independent variable with a convex combination of independent variables. In this talk, we present a completely explicit solution to this problem in two variables.