Extremal solutions of the strong Stieltjes moment problem (Q1917949)

The author shows that a solution of the strong Stieltjes moment problem for the sequence \(\{C_n\); \(n=0, \pm 1, \pm 2,\dots\}\) is a finite positive measure \(\mu\) on \((0, \infty)\) such that \(Cn=\int^\infty_0 t^nd\mu(t)\) for all \(n\), while a solution of the strong Hamburger moment problem for the same sequence is a finite measure \(\mu\) on \((-\infty, \infty)\) such that \(C_n=\int^\infty_{-\infty} t^nd \mu(t)\) for all \(n\). When the Hamburger problem is indeterminate, there exists a one-to-one correspondence between all solutions \(\mu\) and all Nevanlinna functions \(\varphi\), the constant \(\infty\) included. The correspondence is given by \[ F\mu(z)= - {\alpha(z) \varphi(z) - \gamma(z) \over \beta(z) \varphi (z)-\delta(z)}, \] where \(\alpha, \beta, \gamma, \delta\) are certain functions holomorphic in \(\mathbb{C} \backslash\{0\}\).