Geodesic
Nonconstructive proofs of the Beurling--Malliavin theorem on the radius of completeness, and nonuniqueness theorems for entire functions
Izvestiya. Mathematics , Tome 45 (1995) no. 1, pp. 125-149

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Two new methods for proving the Beurling–Malliavin theorem on the radius of completeness are given. Development of the first method allows one to obtain new sufficient conditions for a sequence $\Lambda=\{\lambda_n\}\subset\mathbf C$ to be a set of nonuniqueness for a wide class of weighted spaces of entire functions, and development of the second gives conditions for this property to be preserved under small displacements of the points $\lambda_n$. 
@article{IM2_1995_45_1_a5,
     author = {B. N. Khabibullin},
     title = {Nonconstructive proofs of the {Beurling--Malliavin} theorem on the radius of completeness, and nonuniqueness theorems for entire functions},
     journal = {Izvestiya. Mathematics },
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     publisher = {mathdoc},
     volume = {45},
     number = {1},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1995_45_1_a5/}
}
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B. N. Khabibullin. Nonconstructive proofs of the Beurling--Malliavin theorem on the radius of completeness, and nonuniqueness theorems for entire functions. Izvestiya. Mathematics , Tome 45 (1995) no. 1, pp. 125-149. http://geodesic.mathdoc.fr/item/IM2_1995_45_1_a5/