On the Nevanlinna–Pick interpolation problem in multiply connected domains

V. P. Vinnikov   ; S. I. Fedorov

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 15, Tome 254 (1998), pp. 5-27

Résumé

We simplify and strengthen Abrahamse's result on the Nevanlinna–Pick interpolation problem in a finitely connected planar domain, according to which the problem has a solution if and only if the Pick matrices associated with character-automorphic Hardy spaces are positive semidefinite for all characters in $\mathbb R^ {n-1}/\mathbb Z^{n-1}$, where $n$ is the connectivity of the domain. The main aim of the paper is to reduce the indicated procedure (verification of the positive semidefiniteness) for the entire real $(n-1)$-torus $\mathbb R^{n-1}/\mathbb Z^{n-1}$ to a part of it, whose dimension is, possibly, less than $n-1$.