Geodesic
The Fourier transform of the characteristic function of a set, vanishing on an interval
Sbornik. Mathematics, Tome 45 (1983) no. 3, pp. 397-410 
P. P. Kargaev. The Fourier transform of the characteristic function of a set, vanishing on an interval. Sbornik. Mathematics, Tome 45 (1983) no. 3, pp. 397-410. http://geodesic.mathdoc.fr/item/SM_1983_45_3_a4/
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     title = {The {Fourier} transform of the characteristic function of a~set, vanishing on an~interval},
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Voir la notice de l'article provenant de la source Math-Net.Ru

A set $E$ with $0<\operatorname{mes}E<+\infty$ is constructed for which the Fourier transform of its characteristic function vanishes on an interval. The set is the union of a sequence of intervals whose lengths can be estimated asymptotically above and below. The construction is based on an infinite-dimensional version of the implicit function theorem. Bibiography: 6 titles.

[1] Sapogov N. A., “O preobrazovanii Fure indikatora mnozhestva konechnoi lebegovoi mery v $\mathbf R^n$”, 3an. nauchn. sem. LOMI, 81 (1978), 73

[2] Beurling A., Lectures on quasi-analiticity, A. M. S. Summer Institute, Stanford, 1961

[3] Sapogov N. A., “Ob odnoi probleme edinstvennosti dlya konechnykh mer v evklidovykh prostranstvakh”, Zap. nauchn. sem. LOMI, 41 (1974), 3–13 | MR | Zbl

[4] Kartan A., Differentsialnoe ischislenie, differentsialnye formy, Mir, M., 1971 | MR | Zbl

[5] Mandelbroit S., Ryady Dirikhle, printsipy i metody, Mir, M., 1973 | Zbl

[6] Lukach E., Kharakteristicheskie funktsii, Nauka, M., 1979 | MR | Zbl