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.
Categories: math.CV math.FA