Let σ > 0. The symbol B ∞ σ denotes the space of all entire functions of exponential type not exceeding σ that are bounded on the real axis. Various exact descriptions of uniqueness sequences for the Bernstein spaces B ∞ σ are given in terms of σ and the Poisson and Hilbert transformations. These descriptions lead to completeness criteria for systems of exponentials (up to one or two members) in various classical function spaces on an interval (closed or open) of length d.

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