Hamburger moment problem

In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence { m_n : n = 1, 2, 3, ... }, does there exist a positive Borel measure μ on the real line such that

In other words, an affirmative answer to the problem means that { m_n : n = 0, 1, 2, ... } is the sequence of moments of some positive Borel measure μ.

The Stieltjes moment problem, Vorobyev moment problem, and the Hausdorff moment problem are similar but replace the real line by ${\displaystyle [0,+\infty )}$ (Stieltjes and Vorobyev; but Vorobyev formulates the problem in the terms of matrix theory), or a bounded interval (Hausdorff).

The Hamburger moment problem is solvable (that is, {m_n} is a sequence of moments) if and only if the corresponding Hankel kernel on the nonnegative integers

is positive definite, i.e.,

for an arbitrary sequence {c_j}_{j ≥ 0} of complex numbers with finite support (i.e. c_j = 0 except for finitely many values of j).

For the "only if" part of the claims simply note that

which is non-negative if ${\displaystyle \mu }$ is non-negative.

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