Algebraic description of the finite Stieltjes moment problem

Abstract

The Stieltjes problem of moments seeks for a nondecreasing positive distribution function μ(λ) on the semi-axis [0,+∞) so that its moments match a given infinite sequence of positive real numbers m0,m1,…. In his seminal paper Investigations on continued fractions published in 1894 Stieltjes gave a complete solution including the conditions for the existence and uniqueness in relation to his main goal, the convergence theory of continued fractions.One can also reformulate the Stieltjes problem of moments as looking for a sequence of positive distribution functionsμ(1)(λ),μ(2)(λ),…, where the nth distribution function has n points of increase and m0,m1,…,m2n−1 represent its (first) 2n moments, i.e., as the sequence of the finite Stieltjes moment problems. This view can be linked to iterative solution of (large) linear algebraic systems. Providing that m0,m1,…are moments of some linear, self-adjoint and coercive operator A on a Hilbert space with respect to a given vector f, the finite Stieltjes moment problems determine the iterations of the conjugate gradient method applied for solving Au=f, and vice versa. Here the existence and uniqueness is guaranteed by the properties of the operator A (reformulation for finite sequences, matrices and finite vectors is obvious).This fundamental link raises a question on how the solution of the finite Stieltjes moment problem can be described purely algebraically. This has motivated the presented exposition built upon ideas published previously by several authors. Since the description uses matrices of moments, it is not intended for numerical computations.