Quantitative estimates of sampling constants in model spaces

Andreas Hartmann (1) , Philippe Jaming (1) , Karim Kellay (1)

1 IMB - Institut de Mathématiques de Bordeaux

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 theorem for model spaces associated with bounded derivative inner functions. Considering meromorphic inner functions allows us to introduce a new geometric density condition, in terms of which the sampling sets are completely characterized. This, in comparison to Volberg's characterization of sampling measures in terms of harmonic measure, enables us to obtain explicit estimates on the sampling constants. The methods combine Baranov-Bernstein inequalities, reverse Carleson measures and Remez inequalities .

Mots clés: Model space; Bernstein inequalities; sampling; reverse Carleson measure

Domaines: Variables complexes [math.CV]

Andreas Hartmann, Philippe Jaming, Karim Kellay. Quantitative estimates of sampling constants in model spaces. American Journal of Mathematics, 2020, 142 (4), pp.1301-1326. ⟨hal-01566472⟩