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

Abstract. In this paper we study interpolating sequences for two related spaces of holomorphic functions in the unit ball of C n , n > 1. We first give density conditions for a sequence to be interpolating for the class A −∞ of holomorphic functions with polynomial growth. The sufficient condition is formally identical to the characterizing condition in dimension 1, whereas the necessary one goes along the lines of the results given by Li and Taylor for some spaces of entire functions. In the second part of the paper we show that a density condition, which for n = 1 coincides with the characterizing condition given by Seip, is sufficient for interpolation in the (weighted) Bergman space.

We study a generalized interpolation problem for the space H ∞ (B 2 ) of bounded homomorphic functions in the ball B 2 . A sequence Z = {zn} of B 2 is an interpolating sequence of order 1 if for all sequence of values wn satisfying conditions of order 1 (that is discrete derivatives in the pseudohyperbolic metric are bounded) there exists a function f ∈ H ∞ (B 2 ) such that f (zn) = wn. These sequences are characterized as unions of 3 free interpolating sequences for H ∞ (B 2 ) such that all triplets of Z made of 3 nearby points have to define an angle uniformly bounded below (in an appropriate sense). Also, we give a multiple interpolation result (interpolation of values and derivatives).

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