Completeness in $L^1 (\mathbb R)$ of discrete translates
Bruna , Joaquim ; Olevskii , Alexander ; Ulanovskii , Alexander
Rev. Mat. Iberoamericana, Tome 22 (2006) no. 2, p. 1-16 / Harvested from Project Euclid
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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.
Publié le : 2006-05-15
Classification:  discrete translates,  generator,  Beurling-Malliavin density,  uniqueness sets,  Bernstein classes,  42A65,  42C30,  30D20,  30D15 
@article{1148492174,
     author = {Bruna ,  Joaquim and Olevskii ,  Alexander and Ulanovskii ,  Alexander},
     title = {Completeness in $L^1 (\mathbb R)$ of discrete translates},
     journal = {Rev. Mat. Iberoamericana},
     volume = {22},
     number = {2},
     year = {2006},
     pages = { 1-16},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1148492174}
}

Bruna ,  Joaquim; Olevskii ,  Alexander; Ulanovskii ,  Alexander. Completeness in $L^1 (\mathbb R)$ of discrete translates. Rev. Mat. Iberoamericana, Tome 22 (2006) no. 2, pp.  1-16. http://gdmltest.u-ga.fr/item/1148492174/