Multivariable approximate Carleman-type theorems for complex measures

We prove a multivariable approximate Carleman theorem on the determination of complex measures on ${\mathbb{R}}^n$ and ${\mathbb{R}}^n_+$ by their moments. This is achieved by means of a multivariable Denjoy--Carleman maximum principle for quasi-analytic functions of several variables. As an application, we obtain a discrete Phragm\'{e}n--Lindel\"{o}f-type theorem for analytic functions on ${\mathbb{C}}_+^n$.