Truncated moment problems in the class of generalized Nevanlinna functions

TL;DR

This paper investigates truncated moment problems within generalized Nevanlinna functions, establishing solvability criteria, parametrizations, and analyzing the complex case of degenerate Hankel matrices, revealing potential for infinitely many solutions.

Contribution

It provides new solvability criteria and solution parametrizations for truncated moment problems in generalized Nevanlinna functions, especially addressing degenerate Hankel matrices and indefinite cases.

Findings

Degenerate moment problems can have infinitely many solutions.

A step-by-step Schur algorithm is adapted for indefinite cases.

Complete solution descriptions are obtained for both even and odd problems.

Abstract

Truncated moment problems in the class of generalized Nevanlinna functions are investigated. General solvability criteria will be established, covering both the even and odd problems, including complete parametrizations of solutions. The main new results concern the case where the corresponding Hankel matrix of moments is degenerate. One of the new effects which reveals in the indefinite case is that the degenerated moment problem may have infinitely many solutions. However, with a careful application of an indefinite analogue of a step-by-step Schur algorithm a complete description of the set of solutions will be obtained.