Sampling, interpolation and Riesz bases in small Fock spaces

Anton Baranov (1) , André Dumont (2) , Andreas Hartmann (3) , Karim Kellay (3)

1 Department of Mathematics and Mechanics
2 I2M - Institut de Mathématiques de Marseille
3 IMB - Institut de Mathématiques de Bordeaux

We give a complete description of Riesz bases of reproducing kernels in small Fock spaces. This characterization is in the spirit of the well known Kadets--Ingham 1/4 theorem for Paley--Wiener spaces. Contrarily to the situation in Paley--Wiener spaces, a link can be established between Riesz bases in the Hilbert case and corresponding complete interpolating sequences in small Fock spaces with associated uniform norm. These results allow to show that if a sequence has a density stricly different from the critical one then either it can be completed or reduced to a complete interpolating sequence. In particular, this allows to give necessary and sufficient conditions for interpolation or sampling in terms of densities.

Mots clés: Riesz bases; small Fock spaces; Sampling; interpolation; de Branges spaces; Bari's theorem

Domaines: Variables complexes [math.CV]

Anton Baranov, André Dumont, Andreas Hartmann, Karim Kellay. Sampling, interpolation and Riesz bases in small Fock spaces. Journal de Mathématiques Pures et Appliquées, 2015, 103 (6), pp.1358-1389. ⟨hal-01001761v2⟩