An Upper Bound for the Nevanlinna Matrix of an Indeterminate Moment Sequence

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 into a discrete vector-valued differential equation. 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.