A generalization of Carleman's uniqueness theorem and a discrete Phragmén-Lindelöf theorem

Let $d\mu \geq 0$ be a Borel measure on $[0,\infty )$ and $A_{n}=\int \limits _{0}^{\infty }t^{n}d\mu (t) < \infty ~~(n=0,1,2,...)$ be its moments. T. Carleman found sharp conditions on the magnitude of $\{A_{n}\}_{0}^{\infty }$ for $d\mu$ to be uniquely determined by its moments. We show that the same conditions ensure a stronger property: if $A_{n}' =\int \limits _{0}^{\infty }t^{n} d\mu _{1} (t)$ are the moments of another measure, $d\mu _{1} \geq 0,$ with $\limsup \limits _{n\to \infty } |A_{n}-A_{n}'|^{\frac{1}{n}}=\rho <\infty ,$ then the measure $d\mu -d\mu _{1}$ is supported on the interval $[0,\rho ].$ This result generalizes both the Carleman theorem and a theorem of J. Mikusi\'{n}ski. We also present an application of this result by establishing a discrete version of a Phragmén-Lindelöf theorem.