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Published online by Cambridge University Press: 14 September 2020
The moment problem is the characterization of those real sequences that can appear as moment sequences together with the problem of recovering a positive measure from its moments. So given a sequence of real numbers (mn)n we wish to find out if there exists a positive measure μ such that
mn=∫ℝxndμ(x) , n≥0, (11.1.1)
and in the affirmative case to find all such measures. This is the Hamburger moment problem if there is no restriction imposed on the support of μ. One usually normalizes the problem by requiring m0=1. Hamburger proved that a sequence of real numbers (mn)n is a Hamburger moment sequence if and only if the sequence is positive definite in the sense that
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