Interpolation and random interpolation in de Branges-Rovnyak spaces

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).

💡 Research Summary

This paper presents a comprehensive study of interpolation problems in de Branges-Rovnyak spaces H(b), focusing on the case where the defining function b is a non-extreme rational function in the unit ball of H∞. The central objective is to characterize sequences in the unit disk D that serve as interpolating sequences for these spaces, both in the universal sense (interpolating all ℓ2 sequences with bounded linear operators) and in the multiplier sense (interpolating all ℓ∞ sequences with functions from the multiplier algebra M(H(b))).

The analysis is built upon a precise structural description of H(b) spaces when b is rational and non-extreme. The key tool is Theorem 1.1, which states that such an H(b) space decomposes into a direct sum: H(b) =