Nevanlinna-Pick interpolation in the right half-plane

TL;DR

This paper investigates Nevanlinna-Pick interpolation in the right half-plane, revealing limitations of the Szeg"o-Dirichlet kernel's powers and providing a network realization formula for the half-plane.

Contribution

It demonstrates that positive integer powers of the Szeg"o-Dirichlet kernel lack the 2-point scalar Pick property and introduces a network realization formula for the right half-plane.

Findings

No positive integer powers of the kernel have 2-point scalar Pick property

A network realization formula for the right half-plane is derived

Insights into the structure of kernels in complex analysis

Abstract

The Szeg\"o-Dirichlet kernel of the right half-plane $\mathbb H_{1/2}$ is given by ${\varkappa}(s, u) = \zeta(s+\overline{u}),$ $s, u \in \mathbb H_{1/2},$ where $\zeta$ denotes the Riemann zeta function. We show that none of the positive integer powers of $\varkappa$ has $2$-point scalar Pick property. Nevertheless, a network realization formula for the right half-plane $\mathbb H_0$ is obtained.