Quantitative estimates of sampling constants in model spaces

TL;DR

This paper provides explicit quantitative estimates for sampling sets in model spaces linked to meromorphic inner functions, extending classical theorems and introducing a new geometric density condition for characterizing sampling sets.

Contribution

It extends the Logvinenko-Sereda-Panejah theorem to model spaces with meromorphic inner functions and introduces a new geometric density condition for sampling sets.

Findings

Includes Kovrijkine's optimal sampling constants for Paley-Wiener spaces.

Extends Dyakonov's theorem to model spaces with bounded derivative inner functions.

Provides explicit estimates on sampling constants using geometric density conditions.

Abstract

We establish quantitative estimates for sampling (dominating) sets in model spaces associated with meromorphic inner functions, i.e. those corresponding to de Branges spaces. Our results encompass the Logvinenko-Sereda-Panejah (LSP) Theorem including Kovrijkine's optimal sampling constants for Paley-Wiener spaces. It also extends Dyakonov's LSP theoremfor model spaces associated with bounded derivative inner functions. Considering meromorphic inner functions allows us tointroduce a new geometric density condition, in terms of which the sampling sets are completely characterized. This, incomparison to Volberg's characterization of sampling measures in terms of harmonic measure, enables us to obtain explicitestimates on the sampling constants. The methods combine Baranov-Bernstein inequalities, reverse Carleson measures andRemez inequalities .