Hamburger moment problem

The Hamburger moment problem is solvable (that is, is a sequence of moments) if and only if the corresponding Hankel kernel on the nonnegative integers : A = \left(\begin{matrix} m_0 & m_1 & m_2 & \cdots \\ m_1 & m_2 & m_3 & \cdots \\ m_2 & m_3 & m_4 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{matrix}\right) is positive definite, i.e., : \sum_{j,k\ge0}m_{j+k}c_j\overline{c_k}\ge0 for every arbitrary sequence of complex numbers that are finitary (i.e., except for finitely many values of ). For the "only if" part of the claims simply note that : \sum_{j,k\ge0}m_{j+k}c_j \overline{c_k} = \int_{-\infty}^\infty \left|\sum_{j\geq 0} c_j x^j\right|^2\,d \mu(x) , which is non-negative if \mu is non-negative. We sketch an argument for the converse. Let be the nonnegative integers and denote the family of complex valued sequences with finitary support. The positive Hankel kernel induces a (possibly degenerate) sesquilinear product on the family of complex-valued sequences with finite support. This in turn gives a Hilbert space :(\mathcal{H}, \langle\; , \; \rangle) whose typical element is an equivalence class denoted by . Let be the element in defined by . One notices that :\langle [e_{n+1}], [e_m] \rangle = A_{m,n+1} = m_{m+n+1} = \langle [e_n], [e_{m+1}]\rangle. Therefore, the shift operator on \mathcal{H}, with , is symmetric. On the other hand, the desired expression :m_n = \int_{-\infty}^\infty x^n\,d \mu(x) suggests that is the spectral measure of a self-adjoint operator. (More precisely stated, is the spectral measure for an operator \overline{T} defined below and the vector [1], ). If we can find a "function model" such that the symmetric operator is multiplication by, then the spectral resolution of a self-adjoint extension of 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 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 \overline{T} be one of them and μ be its spectral measure. So :\langle \overline{T}^n [1], [1] \rangle = \int x^n d \mu(x). On the other hand, : \langle \overline{T}^n [1], [1] \rangle = \langle T^n [e_0], [e_0] \rangle = m_n. 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 :\Delta_n = \left[\begin{matrix} m_0 & m_1 & m_2 & \cdots & m_{n} \\ m_1 & m_2 & m_3 & \cdots & m_{n+1} \\ m_2 & m_3 & m_4 & \cdots & m_{n+2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ m_{n} & m_{n+1} & m_{n+2} & \cdots & m_{2n} \end{matrix}\right]. Positivity of means that, for each , . If , for some , then :(\mathcal{H}, \langle \; , \; \rangle) is finite-dimensional and is self-adjoint. So in this case the solution to the Hamburger moment problem is unique and , being the spectral measure of , has finite support. More generally, the solution is unique if there are constants and such that, for all , . This follows from the more general Carleman's condition. There are examples where the solution is not unique; see e.g. == Polynomials ==

The Hamburger moment problem is intimately related to orthogonal polynomials on the real line. That is, assume \{m_{n}\}_{n \in \mathbb{N}_{0}} is the moment sequence of some positive measure \mu on \mathbb{R}. Then for any polynomial p(x) = \sum_{j=0}^{n} a_j x^j \in \mathbb{R}, it holds that \int p(x)^2 d\mu = \int\left( \sum_{j,k=0}^{n} a_j a_k x^{j+k} \right) d\mu = \sum_{j,k=0}^{n} a_j a_k m_{j+k} \geq 0, \quad \forall n\in \mathbb{N}_{0}, such that the Hankel matrix is positive semidefinite. This is a necessary condition for a sequence to be a moment sequence and a sufficient condition for the existence of a positive measure. The Gram–Schmidt procedure gives a basis of orthogonal polynomials in which the operator: \overline{T} 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 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 :m(t) = \sum_{n=0}m_n\frac{t^n}{n!}, provided that this function converges. == Notes ==