Geodesic
On absolute convergence of Fourier series of almost periodic functions with sparse spectrum
Sbornik. Mathematics, Tome 10 (1970) no. 1, pp. 37-49 
E. A. Bredikhina. On absolute convergence of Fourier series of almost periodic functions with sparse spectrum. Sbornik. Mathematics, Tome 10 (1970) no. 1, pp. 37-49. http://geodesic.mathdoc.fr/item/SM_1970_10_1_a2/
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     author = {E. A. Bredikhina},
     title = {On absolute convergence of {Fourier} series of almost periodic functions with sparse spectrum},
     journal = {Sbornik. Mathematics},
     pages = {37--49},
     year = {1970},
     volume = {10},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1970_10_1_a2/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The paper contains inequalities for the absolute value of the Fourier coefficients of functions almost periodic in the sense of Stepanov ($S$-a.p. functions) having sparse spectrum, in a sense which we define. In the particular case in which the spectrum has a single limit point at infinity, we obtain generalizations of Theorem 1 of Chao Jai-arng (RZhMat., 1967, 10B123) and Theorem 1 of Hsieh Ting-fan (RZhMat., 1967, 11B102), proved for $2\pi$-periodic functions. The case in which the spectrum has a single limit point is considered. The results are then extended to the case of $S$-a.p. functions whose spectrum has a finite or countable number of isolated limit points. It is indicated how the results may be used to give sufficient conditions for absolute convergence for the Fourier series of $S$-a.p. functions. Bibliography: 14 titles.

[1] B. M. Levitan, Pochti periodicheskie funktsii, Gostekhizdat, Moskva, 1953

[2] I. P. Natanson, Teoriya funktsii veschestvennoi peremennoi, Fizmatgiz, Moskva, 1957 | MR

[3] Xie Ting-Fan, “On lacunary Fourier series. I”, Chinese Math., 5:2 (1964), 340–345 | MR

[4] Xie Ting-Fan, “On lacunary Fourier series. II”, Chinese Math., 8:4 (1966), 534–537 | MR | Zbl

[5] Chao Jia Arang, “On Fourier series with gaps”, Proc Japan Acad., 42:4 (1966), 308–312 | DOI | MR | Zbl

[6] J.-P. Kahane, “Lacunary Taylor and Fourier series”, Bull. Amer. Math. Soc., 70:2 (1964), 199–213 | DOI | MR | Zbl

[7] M. Izumi, S. Izumi, “On lacunary Fourier series”, Proc. Japan Acad., 41:8 (1965), 648–651 | DOI | MR | Zbl

[8] E. A. Bredikhina, “O priblizhenii pochti periodicheskikh funktsii s ogranichennym spektrom”, Matem. sb., 56(98) (1962), 59–76 | Zbl

[9] E. A. Bredikhina, “O priblizhenii pochti periodicheskikh funktsii Stepanova”, DAN SSSR, 164:2 (1965), 255–258 | Zbl

[10] N. I. Akhiezer, Lektsii po teorii approksimatsii, Gostekhizdat, Moskva–Leningrad, 1947 | Zbl

[11] E. A. Bredikhina, “K teoreme S. N. Bernshteina o nailuchshem priblizhenii nepreryvnykh funktsii tselymi funktsiyami dannoi stepeni”, Izv. VUZov, Matematika, 1961, no. 6, 3–7 | Zbl

[12] E. A. Bredikhina, “O nailuchshikh priblizheniyakh pochti-periodicheskikh funktsii, pokazateli Fure kotorykh ne imeyut predelnykh tochek na konechnom rasstoyanii”, Trudy Kuibyshevsk. aviats. in-ta, IV (1958), 15–23

[13] Chen Nai-dun, “O kharakteristikakh strukturnykh svoistv ravnomernykh pochti-periodicheskikh funktsii odnogo klassa”, Scientia sinica, 13:2 (1964), 185–192

[14] E. A. Bredikhina, “O skhodimosti ryadov Fure pochti periodicheskikh funktsii”, DAN SSSR, 160:2 (1965), 259–262 | Zbl