The Classical Moment Problem And Some Related Questions In Analysis -

For the Hamburger problem, this condition is also sufficient (a theorem of Hamburger, 1920): A sequence $(m_n)$ is a Hamburger moment sequence if and only if the Hankel matrix is positive semidefinite.

For the Hausdorff problem (support in $[0,1]$), the condition becomes that the sequence is : the forward differences alternate in sign. Specifically, $\Delta^k m_n \ge 0$ for all $n,k\ge 0$, where $\Delta m_n = m_n+1 - m_n$. 3. Uniqueness: The Problem of Determinacy Even if a moment sequence exists, the measure might not be unique. This is the most subtle part of the theory. For the Hamburger problem, this condition is also

For the Stieltjes problem (support on $[0,\infty)$), we need an extra condition: both the Hankel matrix of $(m_n)$ and the shifted Hankel matrix of $(m_n+1)$ must be positive semidefinite. For the Stieltjes problem (support on $[0,\infty)$), we

$$ m_n = \int_\mathbbR x^n , d\mu(x) $$