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

Abstract

Abstract. We give a characterization for the existence of a holomorphic interpolant on the unit polydisc D n , n ≥ 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 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.