Hamburger moment problem

Remarks

The existence question is solved. Hamburger proved that such a measure exists if and only if, for every $n \ge 0$, the Hankel matrix $(m_{j+k})_{j,k=0}^{n}$ is positive semidefinite. Equivalently, $\sum_{j,k=0}^{n} m_{j+k} c_j c_k \ge 0$ for all $n$ and all real numbers $c_0, \ldots, c_n$, since this sum is $\int \left( \sum_j c_j x^j \right)^2 d\mu$ when $\mu$ exists. Unlike the Hausdorff moment problem on $[0, 1]$, the measure need not be unique. The moment problem is called determinate when it is unique. Carleman's condition $\sum_{n \ge 1} m_{2n}^{-1/(2n)} = \infty$ is sufficient for uniqueness. The moments of the standard normal distribution satisfy it, so that distribution is determined by its moments. The log-normal distribution is a classical example whose moments do not determine it.