The three-point Pick-Nevanlinna interpolation problem on the polydisc

Abstract

We give a characterization for the existence of a holomorphic interpolant on the unit polydisc ${\mathbb{D}}^n,$ $n\ge 2,$ for prescribed three-point Pick--Nevanlinna data. One of the key steps is a characterization for the existence of an interpolant that is a rational inner function on ${\mathbb{D}}^n.$ The latter reduces the search for a three-point interpolant to finding a single rational inner function that satisfies a type of positivity condition and arises from a polynomial of a very special form. This in turn relies on a pair of results, which are of independent interest, on the factorization of rational inner functions.

12 pages; strengthened the main theorem, added references, and added discussion on some consequences of the main theorem

References in corpus (2)

• The von Neumann inequality for 3x3 matrices
• Three-point Nevanlinna Pick problem in the polydisc