For every set S of finite measure in R we construct a discrete set of real frequencies Λ such that the exponential system {exp(iλt), λ ∈ Λ} is a frame in L 2 (S).

We ask if there exist discrete "universal" sets Λ of given finite density such that every signal f with bounded spectrum of small measure can be recovered from the samples f (λ), λ ∈ Λ. We prove that uniqueness in this problem can be achieved in general situation. On the other hand, for stable reconstruction it is crucial whether the spectrum is compact or dense.

The paper discusses sharp sufficient conditions for interpolation and sampling for functions of n variables with convex spectrum. When n = 1, the classical theorems of Ingham and Beurling state that the critical values in the estimates from above (from below) for the distances between interpolation (sampling) nodes are the same. This is no longer true for n > 1. While the critical value for sampling sets remains constant, the one for interpolation grows linearly with the dimension. R n e −it·x F (x) dx of functions F ∈ L 2 (R n ) which vanish a.e. outside S. Equipped with the L 2 -norm, P W S is a Hilbert space. Definition 1.2 Let S ⊂ R n be a compact set. The Bernstein space B S consists of all continuous bounded functions on R n which are the Fourier transforms of distributions supported by S.Equipped with the L ∞ −norm, B S is a Banach space. Let S be a closed convex symmetric body in R n . We denote by x S = min{r ≥ 0 : x ∈ rS} the norm generated by S, and by S o := {x ∈ R n : x · y ≤ 1, y ∈ S} the polar body of S.When S is a closed convex symmetric body, the Paley-Wiener and Bernstein spaces admit a simple characterization:

We characterize, in terms of the Beurling-Malliavin density, the discrete spectra Λ ⊂ R for which a generator exists, that is a functionIt is shown that these spectra coincide with the uniqueness sets for certain analytic classes. We also present examples of discrete spectra Λ ⊂ R which do not admit a single generator while they admit a pair of generators.

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