Interpolation problems on the spectrum of H ∞

Abstract

We characterize those sequences (xn) in the spectrum of H∞ whose Nevanlinna–Pick interpolation problems admit thin Blaschke products as solutions. We also study under which conditions there is a Blaschke product B with prescribed zero-set distribution and solving problems of the form B(x) = fn(x) for every x ∈ P(xn), where P(xn) is the Gleason part associated with the point xn and where (fn) is an arbitrary sequence of functions in the unit ball of H∞. As a corollary we get a new characterization of Carleson–Newman Blaschke products in terms of bounded universal functions, a result first proved by Gallardo and Gorkin.