I am considering the Stieljes moment problem (https://en.wikipedia.org/wiki/Stieltjes_moment_problem), and its solution for a special class of moment sequences derived from quantum mechanics. One is given a moment sequence $\{m_n\}$ for a measure $d\mu$ with support on the half line. Due to the way the moment sequence is generated, I know all terms $m_n$ are finite. The question is given the moment sequence, perhaps only up to $K$ terms, what can be determined about the associated measure $d\mu$?
Forgive my inaccurate use of distributional terminology: I think that finiteness of all moments implies that the measure $d\mu$ is 'Schwartz', viewed as a distribution (since all moment integrals converge). Letting $d\mu = f(x)dx$, this compactness implies that the Laplace transform should converge for all $s > 0$: $$ \mathcal{L}[f](s)=\int_0^\infty e^{-sx} f(x)dx < \infty$$ One can write the Laplace transform as a power series in the moments.
My questions are twofold:
- Given a complete moment sequence $\{m_n\}_{n=0}^\infty$, one has the Laplace transform of the distribution in question. What invertibility conditions exist for $\mathcal{L}[f](s)$, and what about their uniqueness?
- Given an incomplete moment sequence $\{m_n\}_{n = 0}^K$, what can be said about the distribution?
The goal would be to extract at least an estimation of the functional form of the measure given an incomplete moment sequence, ideally with some control on its error as function of $K$. However I am not sure what truncation of the sum does the analyticity properties of the Laplace transform, and how it affects invertibility.