TY - JOUR

T1 - Almost integer translates. Do nice generators exist?

AU - Olevskii, Alexander

AU - Ulanovskii, Alexander

PY - 2004

Y1 - 2004

N2 - It has been shown earlier by the first author that for any nonzero perturbation of the integers λn = n + 0(1), λn ≠ n, there is a generator, that is a function ψ ∈ L2(R) such that the system of translates {ψ(x - λn)} is complete in L2(R). We ask if ψ can be chosen with fast decay. We prove that in general it cannot. On the other hand, if the perturbations are 'quasianalytically small,' than it can, and this decay restriction is sharp. A certain class of complex measures which we call 'shrinkable' is introduced, and it is shown that the zeros sets of such measures do dot admit generators with fast decay.

AB - It has been shown earlier by the first author that for any nonzero perturbation of the integers λn = n + 0(1), λn ≠ n, there is a generator, that is a function ψ ∈ L2(R) such that the system of translates {ψ(x - λn)} is complete in L2(R). We ask if ψ can be chosen with fast decay. We prove that in general it cannot. On the other hand, if the perturbations are 'quasianalytically small,' than it can, and this decay restriction is sharp. A certain class of complex measures which we call 'shrinkable' is introduced, and it is shown that the zeros sets of such measures do dot admit generators with fast decay.

KW - Beurling-Malliavin density

KW - Exponential systems

KW - Generators

KW - Quasi-analyticity

UR - http://www.scopus.com/inward/record.url?scp=2442587450&partnerID=8YFLogxK

U2 - 10.1007/s00041-004-8006-2

DO - 10.1007/s00041-004-8006-2

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AN - SCOPUS:2442587450

SN - 1069-5869

VL - 10

SP - 93

EP - 104

JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

IS - 1

ER -