Abstract
We present an example of a set {\( \wedge \in \mathbb{Z}\)} satisfying the following two conditions: (1) there exists a nonzero positive singular measure on the unit circle {\(\mathbb{T}\)} with spectrum in Λ; (2) if the spectrum of f∈L1 {\(\left( \mathbb{T} \right)\)} is contained in Λ and f vanishes on a set of positive measure, then f=0. Bibliography: 3 titles.
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References
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A. B. Aleksandrov, “Lacunary series and pseudocontinuations. An arthmetical approach,”Algebra Analiz,9, 3–31 (1997).
A. Ostrowski, “Über Singularitäten gewisser mit Lücken behafteten Potenzreihen,”Jber. Dtsch. Math. Ver.,35, 269–280 (1926).
Additional information
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 7–14.
Translated by A. B. Aleksandrov.
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Aleksandrov, A.B. On a uniqueness theorem for functions with a sparse spectrum. J Math Sci 101, 3049–3052 (2000). https://doi.org/10.1007/BF02673729
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DOI: https://doi.org/10.1007/BF02673729