Interpolating and sampling sequences for entire functions

Abstract

We characterise interpolating and sampling sequences for the spaces of entire functions f such that f e −φ ∈ L p (C), p ≥ 1 (and some related weighted classes), where φ is a subharmonic weight whose Laplacian is a doubling measure. The results are expressed in terms of some densities adapted to the metric induced by ∆φ. They generalise previous results by Seip for the case φ(z) = |z| 2 , and by Berndtsson & Ortega-Cerdà and Ortega-Cerdà & Seip for the case when ∆φ is bounded above and below. CONTENTS 1. Introduction 2. Subharmonic functions with doubling Laplacian 2.1. Doubling measures 2.2. Flat weights 2.3. Local behaviour and regularisation of φ 2.4. The multiplier 3. Basic properties of functions in F p φ,ω 3.1. Pointwise estimates 3.2. Hörmander type estimates 3.3. Bergman kernel estimates 3.4. Scaled translations and invariance 3.5. Weak limits. 4. Preliminary properties of sampling and interpolating sequences 4.1. Weak limits and interpolating and sampling sequences 4.2. Non-existence of simultaneously sampling and interpolating sequences

Sampling and interpolation in large Bergman and Fock spaces

“…Recently, N. Marco, X. Massaneda and J. Ortega-Cerdà [18] described sampling and interpolation sets for the Fock spaces A p h (C) for a wide class of h such that h is a doubling measure. These results rely mainly upon the method used by A. Beurling [4] in his work on band-limited functions and on Hörmander-type weighted estimates for the∂ equation.…”

Bergman Kernel Estimates for Generalized Fock Spaces

“…For the generalized Fock spaces, J. Marzo and J. Ortega-Cerdà [9] obtained similar estimates under the hypothesis that ϕ is a subharmonic function whose Laplacian ∆ϕ is a doubling measure (see [8], [9]). In our paper, we prove the size estimates without the doubling condition.…”

Inhomogeneous random zero sets

“…Moreover there is always a regularisation of the measure ∆φ that is locally flat (see [MMO03,Theorem 14]), but we shall not pursue this line of thought.…”

“…Sampling sequences in the Fock spaces we consider have been characterised in terms of a Beurling-type density. The following definitions appear in [MMO03].…”