Hamburger Moment Problem - Characterization

Characterization

The Hamburger moment problem is solvable (that is, {m_n} is a sequence of moments) if and only if the corresponding Hankel kernel on the nonnegative integers

is positive definite, i.e.,

for an arbitrary sequence {c_j}_{j ≥ 0} of complex numbers with finite support (i.e. c_j = 0 except for finitely many values of j).

The "only if" part of the claims can be verified by a direct calculation.

We sketch an argument for the converse. Let Z+ be the nonnegative integers and F₀(Z+) denote the family of complex valued sequences with finite support. The positive Hankel kernel A induces a (possibly degenerate) sesquilinear product on the family of complex valued sequences with finite support. This in turn gives a Hilbert space

whose typical element is an equivalence class denoted by .

Let e_n be the element in F₀(Z+) defined by e_n(m) = δ_nm. One notices that

Therefore the "shift" operator T on, with T =, is symmetric.

On the other hand, the desired expression

suggests that μ is the spectral measure of a self-adjoint operator. If we can find a "function model" such that the symmetric operator T is multiplication by x, then the spectral resolution of a self-adjoint extension of T proves the claim.

A function model is given by the natural isomorphism from F₀(Z+) to the family of polynomials, in one single real variable and complex coefficients: for n ≥ 0, identify e_n with xn. In the model, the operator T is multiplication by x and a densely defined symmetric operator. It can be shown that T always has self-adjoint extensions. Let

be one of them and μ be its spectral measure. So

On the other hand,