The Hamburger moment problem is solvable if and only if the corresponding Hankel kernel on the nonnegative integers is positive definite, i.e., for every arbitrary sequence j ≥ 0 of complex numbers with finite support. For the "only if" part of the claims simply note that which is non-negative if is non-negative. We sketch an argument for the converse. Let Z+ be the nonnegative integers and F0 denote the family of complex valued sequences with finite support. The positive Hankel kernel A induces a 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 en be the element in F0 defined by en = δnm. One notices that Therefore, the "shift" operatorT 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 to the family of polynomials, in one single real variable and complex coefficients: for n ≥ 0, identify en 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, For an alternative proof of the existence that only uses Stieltjes integrals, see also, in particular theorem 3.2.
Uniqueness of solutions
The solutions form a convex set, so the problem has either infinitely many solutions or a unique solution. Consider the × Hankel matrix Positivity of A means that for each n, det ≥ 0. If det = 0, for some n, then is finite-dimensional and T is self-adjoint. So in this case the solution to the Hamburger moment problem is unique and μ, being the spectral measure of T, has finite support. More generally, the solution is unique if there are constants C and D such that for all n, |mn|≤ CDnn!. This follows from the more general Carleman's condition. There are examples where the solution is not unique, see e.g..
Further results
One can see that the Hamburger moment problem is intimately related to orthogonal polynomials on the real line. The Gram–Schmidt procedure gives a basis of orthogonal polynomials in which the operator: has a tridiagonal Jacobi matrix representation. This in turn leads to a tridiagonal model of positive Hankel kernels. An explicit calculation of the Cayley transform of T shows the connection with what is called the Nevanlinna class of analytic functions on the left half plane. Passing to the non-commutative setting, this motivates Krein's formula which parametrizes the extensions of partial isometries. The cumulative distribution function and the probability density function can often be found by applying the inverse Laplace transform to the moment generating function provided that this function converges.
