Pick Interpolation on the Polydisc: Small Families of Sufficient Kernels

Abstract

Abstract. We give a solution to Pick's interpolation problem on the unit polydisc in C n , n ≥ 2, by characterizing all interpolation data that admit a D-valued interpolant, in terms of a family of positive-definite kernels parametrized by a class of polynomials. This uses a duality approach that has been associated with Pick interpolation, together with some approximation theory. Furthermore, we use duality methods to understand the set of points on the n-torus at which the boundary values of a given solution to an extremal interpolation problem are not unimodular.