Functions of classes $\mathcal N_\varkappa^+$

TL;DR

This paper provides an elementary proof of the necessary and sufficient condition for a univariate function to belong to the class \(\\mathcal{N}_\varkappa^+\), clarifying a gap in previous criteria related to the indefinite Stieltjes moment problem.

Contribution

It offers a new, elementary proof of the membership criterion for \(\mathcal{N}_\varkappa^+\) functions, resolving ambiguities in earlier results.

Findings

Established a precise criterion for \(\mathcal{N}_\varkappa^+\) functions

Closed a gap in the 1977 Krein-Langer criterion

Clarified the correct condition stated in 1998 by Langer and Winkler

Abstract

In the present note we give an elementary proof of the necessary and sufficient condition for a univariate function to belong the class $\mathcal N_\varkappa^+$. This class was introduced mainly to deal with the indefinite version of the Stieltjes moment problem (and corresponding $\pi$-Hermitian operators), although it is applicable beyond the original scope. The proof relies on asymptotic analysis of the corresponding Hermitian forms. Our result closes a gap in the criterion given by Krein and Langer in their joint paper of 1977. The correct condition was stated by Langer and Winkler in 1998, although they provided no proper reasoning.