Moment Problem - Existence

Existence

A sequence of numbers m_n is the sequence of moments of a measure μ if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices H_n,

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 m_kn 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.