Hamburger moment problem

In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence { mn : n = 0, 1, 2, 3, ... }, does there exist a positive Borel measure μ (for instance, the cumulative distribution function of a random variable) on the real line such that In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence { mn : n = 0, 1, 2, 3, ... }, does there exist a positive Borel measure μ (for instance, the cumulative distribution function of a random variable) on the real line such that In other words, an affirmative answer to the problem means that { mn : 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 [ 0 , + ∞ ) {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, {mn} is a sequence of moments) if and only if the corresponding Hankel kernel on the nonnegative integers is positive definite, i.e., for every arbitrary sequence {cj}j ≥ 0 of complex numbers with finite support (i.e. cj = 0 except for finitely many values of j).