The Pick-Nevanlinna Interpolation Problem : Complex-analytic Methods in Special Domains

Abstract

The Pick–Nevanlinna interpolation problem, in its fullest generality, is as follows: Given domains D1, D2 in complex Euclidean spaces, and a set f¹ zi; wiº : 1 i N g D1 D2, where zi are distinct and N 2 š+, N 2, find necessary and sufficient conditions for the existence of a holomorphic map F : D1 ! D2 such that F¹ziº = wi, 1 i N. When such a map F exists, we say that F is an interpolant of the data. Of course, this problem is intractable at the above level of generality. However, two special cases of the problem — which we shall study in this thesis — have been of lasting interest: Interpolation from the polydisc to the unit disc. This is the case D1 = „n and D2 = „, where „ denotes the open unit disc in the complex plane and n 2 š+. The problem itself originates with Georg Pick’s well-known theorem (independently discovered by Nevanlinna) for the case n = 1. Much later, Sarason gave another proof of Pick’s result using an operator-theoretic approach, which is very influential. Using this approach for n 2, Agler–McCarthy provided a solution to the problem with the restriction that the interpolant is in the Schur– Agler class. This is notable because, when n = 2, the latter result completely solves the problem for the case D1 = „2; D2 = „. However, Pick’s approach can also be effective for n 2. In this thesis, we give an alternative characterization for the existence of a 3-point interpolant based on Pick’s approach and involving the study of rational inner functions. Cole–Lewis–Wermer lifted Sarason’s approach to uniform algebras — leading to a char-acterization for the existence of an interpolant in terms of the positivity of a large, rather abstractly-defined family of N N matrices. McCullough later refined their result by identifying a smaller family of matrices. The second result of this thesis is in the same vein, namely: it provides a characterization of those data that admit a „n-to-„ interpolant in terms of the positivity of a family of N N matrices parametrized by a class of polynomials. Interpolation from the unit disc to the spectral unit ball. This is the case D1 = „ and D2 = n , where n denotes the set of all n n matrices with spectral radius less than 1. The interest in this arises from problems in Control Theory. Bercovici–Foias–Tannenbaum adapted Sarason’s methods to give a (somewhat hard-to-check) characterization for the existence of an interpolant under a very mild restriction. Later, Agler–Young established a relation between the interpolation problem in the spectral unit ball and that in the symmetrized polydisc — leading to a necessary condition for the existence of an interpolant. Bharali later provided a new inequivalent necessary condition for the existence of an interpolant for any n and N = 2. In this thesis, we shall present a necessary condition for the existence of an interpolant in the case when N = 3. This we shall achieve by adapting Pick’s approach and applying the aforementioned result of Bharali.