Geodesic
Order estimates of the modulus of variation of the sum of a lacunary trigonometric series
Sbornik. Mathematics, Tome 189 (1998) no. 5, pp. 639-656 
A. S. Belov. Order estimates of the modulus of variation of the sum of a lacunary trigonometric series. Sbornik. Mathematics, Tome 189 (1998) no. 5, pp. 639-656. http://geodesic.mathdoc.fr/item/SM_1998_189_5_a0/
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     title = {Order estimates of the~modulus of variation of the~sum of a~lacunary trigonometric series},
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Voir la notice de l'article provenant de la source Math-Net.Ru

We find order estimates for the modulus of variation and the averaged modulus of the sum of a lacunary trigonometric series in terms of its coefficients. These interesting global characteristics of a function and their applications have been studied in papers by Chanturiya, Dolzhenko, Sevast'yanov, Sendov, Popov, and others. Since the sum of a lacunary trigonometric series has frequently been used in the theory of functions to provide an example of a function having one property or another, it is useful to know as much as possible about such a function, especially such global characteristics as the modulus of variation and the averaged modulus. We also give necessary and sufficient conditions for the sum of a lacunary trigonometric series to belong to certain classes of functions defined in terms of these characteristics.

[1] Chanturiya Z. A., “Modul izmeneniya i ego primenenie v teorii ryadov Fure”, Dokl. AN SSSR, 214:1 (1974), 63–66 | MR | Zbl

[2] Sevastyanov E. A., “Kusochno-monotonnaya approksimatsiya i $\Phi$-variatsii”, Anal. Math., 1:2 (1975), 141–164 | DOI | MR | Zbl

[3] Sendov B., Popov V., Usrednennye moduli gladkosti, Mir, M., 1988 | MR

[4] Dolzhenko E. P., Sevastyanov E. A., “O priblizhenii funktsii v khausdorfovoi metrike posredstvom kusochno-monotonnykh (v chastnosti, ratsionalnykh) funktsii”, Matem. sb., 101:4 (1976), 508–541 | MR | Zbl

[5] Sevastyanov E. A., “Srednii modul kolebaniya i kusochno-monotonnaya approksimatsiya”, Matem. zametki, 31:6 (1982), 867–876 | MR

[6] Avdispahic M., “Concepts of generalized bounded variation and the theory of Fourier series”, Internat. J. Math. Math. Sci., 9:2 (1986), 223–244 | DOI | MR | Zbl

[7] Belov A. S., “Lakunarnye trigonometricheskie ryady i funktsii obobschennoi ogranichennoi variatsii”, Soobscheniya AN GSSR, 134:3 (1989), 49–52 | MR

[8] Zigmund A., Trigonometricheskie ryady, T. 1, Mir, M., 1965 | MR

[9] Belov A. S., “On some local properties of the sum of lacunary trigonometric series”, Anal. Math., 14:1 (1988), 65–97 | DOI | MR | Zbl

[10] Belov A. S., “O neravenstve Sidona–Zigmunda v teorii lakunarnykh trigonometricheskikh ryadov”, Matem. zametki, 30:4 (1981), 501–515 | MR | Zbl

[11] Belov A. S., “O summe lakunarnogo ryada”, Tr. MMO, 33, URSS, M., 1975, 107–153 | MR | Zbl