Some Propositions on Generalized Nevanlinna Functions of the Class Nk

Abstract

Some propositions on the generalized Nevanlinna functions are derived. We indicate mainly that (1) the negative inertia index of a Hermitian generalized Loewner matrix generated by a generalized Nevanlinna function in the class Nκ does not exceed κ. This leads to an equivalent definition of a generalized Nevanlinna function; (2) if a generalized Nevanlinna function in the class Nκ has a uniform asymptotic expansion at a real point α or at infinity, then the negative inertia index of the Hankel matrix constructed with the partial coefficients of that asymptotic expansion does not exceed κ. Also, an explicit formula for the negative index of a real rational function is given by using relations among Loewner, Bézout, and Hankel matrices. These results will provide first tools for the solution of the indefinite truncated moment problems together with the multiple Nevanlinna-Pick interpolation problems in the class Nκ based on the so-called Hankel vector approach.