Existence
A sequence of numbers mn is the sequence of moments of a measure μ if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices Hn,
should be positive semi-definite. A condition of similar form is necessary and sufficient for the existence of a measure supported on a given interval .
One way to prove these results is to consider the linear functional that sends a polynomial
to
If mkn are the moments of some measure μ supported on, then evidently
-
φ(P) ≥ 0 for any polynomial P that is non-negative on .
(1)
Vice versa, if (1) holds, one can apply the M. Riesz extension theorem and extend to a functional on the space of continuous functions with compact support C0, so that
-
(2)
such that ƒ ≥ 0 on .
By the Riesz representation theorem, (2) holds iff there exists a measure μ supported on, such that
for every ƒ ∈ C0.
Thus the existence of the measure is equivalent to (1). Using a representation theorem for positive polynomials on, one can reformulate (1) as a condition on Hankel matrices.
See Refs. 1–3. for more details.
Read more about this topic: Moment Problem
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