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Completeness in \(L^1(\mathbb R)\) of discrete translates. (English) Zbl 1104.42019

Summary: We characterize, in terms of the Beurling-Malliavin density, the discrete spectra \(\Lambda\subset\mathbb{R}\), for which a generator exists, that is a function \(\varphi\in L^1(\mathbb{R})\) such that its \(\Lambda\)-translates \(\varphi(x-\lambda)\), \(\lambda\in\Lambda\), span \(L^1(\mathbb{R})\). It is shown that these spectra coincide with the uniqueness sets for certain analytic classes. We also present examples of discrete spectra \(\Lambda\subset\mathbb{R}\) which do not admit a single generator while they admit a pair of generators.

MSC:

42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
42A65 Completeness of sets of functions in one variable harmonic analysis
30D20 Entire functions of one complex variable (general theory)
30D15 Special classes of entire functions of one complex variable and growth estimates
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References:

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