Math 2.0

Integral transforms, operational calculus Moment problems Problem

Moment problem

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Problem

Let KK be a closed subset of ℝ\R and let (mn)n≥0(m_n)_{n \ge 0} be a sequence of real numbers.

  1. Existence: decide whether there is a positive Borel measure μ\mu on ℝ\R with support in KK and the following property. Every xnx^n is μ\mu-integrable and mn=∫xn dμ(x)m_n = \int x^n \, d\mu(x) for all n≥0n \ge 0.
  2. Uniqueness: when such a measure exists, decide whether it is the only one.

The numbers mnm_n are then the moments of μ\mu.

Remarks

The classical cases are named after the set KK. For K=ℝK = \R it is the Hamburger moment problem, for K=[0,∞)K = [0, \infty) the Stieltjes moment problem, and for K=[0,1]K = [0, 1] the Hausdorff moment problem. For these three cases the existence question is solved by positivity conditions on Hankel matrices built from the moments, or on differences of the sequence for [0,1][0, 1]. For compact KK the measure is unique when it exists, since polynomials are dense in the continuous functions on KK. For unbounded KK uniqueness can fail; the log-normal distribution is the standard example. Many variants are studied: finitely many moments (the truncated problem), measures on ℝd\R^d, and the trigonometric problem on the circle.

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Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Moment_problem
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