Inverse of Infinite Hankel Moment Matrices

TL;DR

This paper investigates conditions under which an inverse matrix exists for infinite Hankel matrices associated with indeterminate moment problems, focusing on the property (aci) and its implications for classical and specific cases.

Contribution

It establishes sufficient conditions for the existence of an inverse matrix with property (aci) in indeterminate moment problems and explores its uniqueness and applicability.

Findings

(aci) holds for many classical indeterminate moment problems.

Rapid increase of recurrence coefficients ensures (aci).

Inverse matrix can be highly non-unique in some cases.

Abstract

Let $(s_n)_{n\ge 0}$ denote an indeterminate Hamburger moment sequence and let $\mathcal H=\{s_{m+n}\}$ be the corresponding positive definite Hankel matrix. We consider the question if there exists an infinite symmetric matrix $\mathcal A=\{a_{j,k}\}$, which is an inverse of $\mathcal H$ in the sense that the matrix product $\mathcal A\mathcal H$ is defined by absolutely convergent series and $\mathcal A\mathcal H$ equals the identity matrix $\mathcal I$, a property called (aci). A candidate for $\mathcal A$ is the coefficient matrix of the reproducing kernel of the moment problem, considered as an entire function of two complex variables. We say that the moment problem has property (aci), if (aci) holds for this matrix $\mathcal A$. We show that this is true for many classical indeterminate moment problems but not for the symmetrized version of a cubic birth-and-death process studied by Valent and co-authors. We consider mainly symmetric indeterminate moment problems and give a number of sufficient conditions for (aci) to hold in terms of the recurrence coefficients for the orthonormal polynomials. A sufficient condition is a rapid increase of the recurrence coefficients in the sense that the quotient between consecutive terms is uniformly bounded by a constant strictly smaller than one. We also give a simple example, where (aci) holds, but an inverse matrix of $\mathcal H$ is highly non-unique.