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Moment Problem

The moment problem asks whether a given sequence can be represented as the moments of a positive measure and whether that measure is unique. The support of the measure is part of the problem. For a prescribed subset D of the real line, the required representation is

mu_n=int_Dt^ndmu(t)

for n=0, 1, .... Such a sequence is a moment sequence.

Hausdorff's moment problem, also called the little moment problem, takes D=[0,1]. It asks for a nondecreasing function alpha of bounded variation on this interval such that

mu_n=int_0^1t^ndalpha(t).

Hausdorff (1921ab) obtained necessary and sufficient conditions for this representation.

A positive measure on a bounded closed interval is uniquely determined by all its moments. This need not hold on the entire real line. Favard's theorem gives a positive measure for a three-term recurrence relation of orthogonal polynomials, but does not in general make that measure unique.

See also

Favard's Theorem, Moment, Moment Sequence, Orthogonal Polynomials

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References

Hausdorff, F. "Summationsmethoden und Momentfolgen. I." Math. Z. 9, 74-109, 1921a.Hausdorff, F. "Summationsmethoden und Momentfolgen. II." Math. Z. 9, 280-299, 1921b.Leviatan, D. "A Generalized Moment Problem." Israel J. Math. 5, 97-103, 1967.Widder, D. V. "The Moment Problem." Ch. 3 in The Laplace Transform. Princeton, NJ: Princeton University Press, pp. 100-101, 1941.

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Moment Problem

Cite this as:

Weisstein, Eric W. "Moment Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MomentProblem.html

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