Given a set of positive measure on the circle and a set Λ of integers, one can ask whether is a Riesz sequence in L²().
We consider this question in connection with some arithmetic properties of the set Λ. Improving a result of Bownik and Speegle (2006), we construct a set such that E(Λ) is never a Riesz sequence if Λ contains an arithmetic progression of length N and step with N arbitrarily large. On the other hand, we prove that every set admits a Riesz sequence E(Λ) such that Λ does contain...
We study the “Fourier symmetry” of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are:
(i) A one-side extension of Frostman’s theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of its support;
(ii) A construction of compacts of “critical” size, which support distributions (even pseudo-functions) with anti-analytic part belonging to .
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We characterize, in terms of the Beurling-Malliavin density, the discrete spectra Λ ⊂ R for which a generator exists, that is a function φ ∈ L(R) such that its Λ translates φ(x - λ), λ ∈ Λ, span L(R). It is shown that these spectra coincide with the uniqueness sets for certain analytic clases. 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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