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

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.