Construction of Solutions of the Hamburger–Löwner Mixed Interpolation Problem for Nevanlinna Class Functions

Abstract

By definition, a Nevanlinna class function $\varphi \in \Re$ is holomorphic and has a nonnegative imaginary part in the half-plane Im z > 0. In this chapter we also consider Nevanlinna functions which belong to the subclass $\Re_0 \subset \Re$ such that if $\varphi(z) \in \Re_0, \lim\nolimits_{z\to \infty} (\varphi(z)/z) = 0$, Im z > 0. Then, due to the Riesz–Herglotz theorem,

$$\varphi(z)=\int\limits^\infty_{-\infty}\frac{d\sigma(t)}{t-z}, \quad {\rm Im} \ z>0,$$

where σ(t) is a nondecreasing function such that

$$\int^\infty_{-\infty} (1+t^2)^{-1} \ d\sigma(t) < \infty$$

. Consider the mixed Löwner–Nevanlinna problem [Löw34, KrNu77, Akh65, KaSt66, CuFi91, CuFi96, AdTk00, UrTkFC01, AdAlTk03], see also [AdTk01(a)] and (for the matrix version of the problem) [AdTk01(b)].