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

TL;DR

This paper extends Hamburger's classical results to characterize the asymptotic behavior at infinity of two-variable Pick functions, using Hankel vector moment sequences to analyze non-tangential regularity.

Contribution

It introduces a novel extension of Hamburger's characterization to two-variable Pick functions using Hankel vector moment sequences.

Findings

Extended Hamburger's results to two-variable case.

Characterized non-tangential regularity at infinity.

Connected asymptotic expansions with moment sequences.

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. H. Hamburger in 1921 characterized which sequences $\{\rho_n\} $ can occur. We give an extension of Hamburger's results to Pick functions of two variables.