Interpolation and random interpolation in de Branges-Rovnyak spaces

TL;DR

This paper characterizes universal and multiplier interpolating sequences in de Branges-Rovnyak spaces for non-extreme rational functions, extending results to higher order local Dirichlet spaces and analyzing random sequences with a 0-1 law.

Contribution

It provides new characterizations of interpolating sequences in de Branges-Rovnyak spaces and extends these results to higher order local Dirichlet spaces, including analysis of random sequences.

Findings

Characterization of universal and multiplier interpolating sequences for non-extreme rational functions.

Extension of results to higher order local Dirichlet spaces.

Establishment of a 0-1 law for random interpolating sequences.

Abstract

The aim of this paper is to characterize universal and multiplier interpolating sequences for de Branges-Rovnyak spaces H (b) where the defining function b is a general non-extreme rational function. Our results carry over to recently introduced higher order local Dirichlet spaces and thus generalize previously known results in classical local Dirichlet spaces. In this setting, we also investigate random interpolating sequences with prescribed radii, providing a 0 -1 law. This condition is automatic when b is rational non inner so that we can assume H (b) = M(a). By standard results in functional analysis, the corresponding norms are equivalent. In [18], the authors demonstrated that the decomposition (1) is orthogonal in the metric of M(a).