Hankel vector moment sequences and the non-tangential regularity at infinity of two variable Pick functions

Abstract:

A Pick function of $d$ variables is a holomorphic map from $\Pi ^d$ to $\Pi$, where $\Pi$ is the upper halfplane. Some Pick functions of one variable have an asymptotic expansion at infinity, a power series $\sum _{n=1}^\infty \rho _n z^{-n}$ with real numbers $\rho _n$ that gives an asymptotic expansion on non-tangential approach regions to infinity. In 1921 H. Hamburger characterized which sequences $\{ \rho _n\}$ can occur. We give an extension of Hamburger’s results to Pick functions of two variables.