Stieltjes moment problem

Remarks

The existence question is solved. Such a measure exists if and only if, for every $n \ge 0$, both Hankel matrices $(m_{j+k})_{j,k=0}^{n}$ and $(m_{j+k+1})_{j,k=0}^{n}$ are positive semidefinite. A measure with infinite support exists if and only if all the determinants of these matrices are strictly positive. Necessity is direct: for real $c_0, \ldots, c_n$, the quadratic forms equal $\int p(x)^2 \, d\mu$ and $\int x \, p(x)^2 \, d\mu$ with $p(x) = \sum_j c_j x^j$, and $x \ge 0$ on the support. The solution need not be unique; the log-normal distribution gives a sequence with many solutions. A Stieltjes moment sequence can be determinate as a Stieltjes problem but indeterminate as a Hamburger problem.