ABSTRACT. The Bohnenblust-Hille inequality says that the ℓ 2m m+1 -norm of the coefficients of an m-homogeneous polynomial P on C n is bounded by P ∞ times a constant independent of n, where · ∞ denotes the supremum norm on the polydisc D n . The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be C m for some C > 1. Combining this improved version of the Bohnenblust-Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc D n behaves asymptotically as (log n)/n modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies log n : n a positive integer ≤ N is √ N exp{(−1/ √ 2 + o(1)) √ log N log log N } as N → ∞.

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

We solve the problem of Duffin and Schaeffer (1952) of characterizing those sequences of real frequencies which generate Fourier frames. Equivalently, we characterize the sampling sequences for the Paley-Wiener space. The key step is to connect the problem with de Branges' theory of Hilbert spaces of entire functions. We show that our description of sampling sequences permits us to obtain a classical inequality of H. Landau as a consequence of Pavlov's description of Riesz bases of complex exponentials and the John-Nirenberg theorem. Finally, we discuss how to transform our description into a working condition by relating it to an approximation problem for subharmonic functions. By this approach, we determine the critical growth rate of a nondecreasing function ψ such that the sequence {λ k } k∈Z defined by λ k + ψ(λ k ) = k is sampling.

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We introduce a new notion for the deformation of Gabor systems. Such deformations are in general nonlinear and, in particular, include the standard jitter error and linear deformations of phase space. With this new notion we prove a strong deformation result for Gabor frames and Gabor Riesz sequences that covers the known perturbation and deformation results. Our proof of the deformation theorem requires a new characterization of Gabor frames and Gabor Riesz sequences. It is in the style of Beurling's characterization of sets of sampling for bandlimited functions and extends significantly the known characterization of Gabor frames "without inequalities" from lattices to non-uniform sets.Comment: 31 pages, 2 figure