StieltjesProblem

The Stieltjes problem, named after Thomas Joannes Stieltjes, refers to a class of moment problems in which one seeks a measure supported on the nonnegative real axis from a given sequence of moments. The most common formulation is the Stieltjes moment problem: given a sequence of real numbers m_k (k = 0, 1, 2, ...), determine whether there exists a nonnegative Borel measure μ on [0, ∞) such that m_k = ∫_0^∞ x^k dμ(x) for all k, and if so, describe μ. This problem is a specialized case of the broader Hamburger moment problem (supported on (−∞, ∞)) and is related to, but more restrictive than, the Hausdorff problem on [0,1].

Existence and uniqueness are central questions. A necessary and sufficient condition for the existence of such a measure is that all associated Hankel matrices built from the moments are positive semidefinite. When a measure exists, questions of determinacy arise: the solution μ may be unique (determinate) or there may be infinitely many measures producing the same moment sequence. Determinacy criteria include Carleman-type conditions; for the Stieltjes problem, there are specialized criteria that reflect the support constraint [0, ∞).

Key tools and methods include the Stieltjes transform S(z) = ∫_0^∞ dμ(t)/(z − t), continued fraction representations (Stieltjes continued fractions), and the theory of orthogonal polynomials with respect to μ. These polynomials satisfy a three-term recurrence encoded in a Jacobi matrix, linking the moment problem to spectral theory and numerical quadrature.

Applications span probability, statistical inference, signal processing, numerical analysis, and mathematical physics, where one often seeks to reconstruct distributions or spectral measures from moments or transform data.