Sampling Sets for the Nevanlinna class

Abstract

We propose a definition of sampling set for the Nevanlinna class in the disk, i.e. a subset of the disk such that the analogue of the norm of a function in the Nevanlinna class can be recovered only from its values on the subset. We show it is equivalent with the notion of determination set for the same class, that is, subsets such that any function in the class which is bounded on the subset must be bounded everywhere (by the same bound, in fact); and more restrictive than the condition of being a determination set for the class of differences of positive harmonic functions (which had been studied in particular by Hayman, Lyons, and Gardiner). We give sufficient conditions for sampling to hold, as well as necessary conditions (different but closely related), and show that in the case of certain (natural) regular sets, the necessary and sufficient conditions coincide and characterize the sampling property in a numerically precise way. In particular, we observe a remarkable agreement with the results of Joaquim Ortega-Cerdà and Kristian Seip about "champagne subdomains", the complement in the unit disk of a union of hyperbolic disks centered on a maximal hyperbolically separated subsequence, the radii of which decrease uniformly as they approach the unit circle (arXiv:math.CV/0305075). Our methods involve a careful analysis of the decrease of the modulus of a Blaschke product.