An upper bound for the Nevanlinna matrix of an indeterminate moment sequence

TL;DR

This paper establishes an explicit upper bound on the growth of the Nevanlinna matrix associated with indeterminate Hamburger moment problems, linking it to the parameters of a corresponding canonical system.

Contribution

It provides a new explicit bound for the Nevanlinna matrix growth in indeterminate moment problems, connecting it to the canonical system parameters and demonstrating sharpness in well-behaved cases.

Findings

Bound for the growth of the Nevanlinna matrix derived

Explicit evaluation of the bound in most cases possible

Bound matches actual growth for well-behaved parameters

Abstract

The solutions of an indeterminate Hamburger moment problem can be parameterised using the Nevanlinna matrix of the problem. The entries of this matrix are entire functions of minimal exponential type, and any growth less than that can occur. An indeterminate moment problem can be considered as a canonical system in limit circle case by rewriting the three-term recurrence of the problem to a first order vector-valued recurrence. We give a bound for the growth of the Nevanlinna matrix in terms of the parameters of this canonical system. In most situations this bound can be evaluated explicitly. It is sharp in the sense that for well-behaved parameters it coincides with the actual growth of the Nevanlinna matrix up to multiplicative constants.