Geodesic
Convergence of series of Fourier coefficients for multiplicative convolutions
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2008), pp. 27-39.

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We study the convergence of series with Fourier–Vilenkin coefficients for functions represented as multiplicative convolutions. In the trigonometric case similar results are obtained by C. Onneweer, M. Izumi, and S. Izumi. Moreover, we consider certain analogs of I. Hirshman and W. Rudin transformations of Fourier coefficients. Some results are proved to be unimprovable in a certain sense.
Keywords: absolute convergence, multiplicative systems
Mots-clés : multiplicative convolution. 
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     author = {S. S. Volosivets},
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S. S. Volosivets. Convergence of series of Fourier coefficients for multiplicative convolutions. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2008), pp. 27-39. http://geodesic.mathdoc.fr/item/IVM_2008_11_a2/

[1] Golubov B. I., Efimov A. V., Skvortsov V. A., Ryady i preobrazovaniya Uolsha: Teoriya i primeneniya, Nauka, M., 1987, 344 pp. | MR | Zbl

[2] Schipp F., Wade W., Simon P., Walsh series, Akad. Kiado, Budapest, 1990, 560 pp.

[3] Chen M. T., “The absolute convergence of Fourier series”, Duke Math. J., 9:4 (1942), 803–810 | DOI | MR | Zbl

[4] Izumi M., Izumi S.-I., “Absolute convergence of Fourier series of convolution functions”, J. Approx. Theory, 1:1 (1968), 103–109 | DOI | MR | Zbl

[5] Onneweer C. W., “On absolutely convergent Fourier series”, Arkiv Mat., 12:1 (1974), 51–58 | DOI | MR | Zbl

[6] Onneweer C. W., “Absolute convergence of Fourier series on certain groups”, Duke Math. J., 39:4 (1972), 599–610 | DOI | MR

[7] Onneweer C. W., “Absolute convergence of Fourier series on certain groups, II”, Duke Math. J., 41:3 (1974), 679–688 | DOI | MR | Zbl

[8] Agaev G. N., Vilenkin N. Ya., Dzhafarli G. M., Rubinshtein A. I., Multiplikativnye sistemy funktsii i garmonicheskii analiz na nul-mernykh gruppakh, ELM, Baku, 1981, 180 pp.

[9] Zigmund A., Trigonometricheskie ryady, T. 1, Mir, M., 1965, 616 pp.

[10] Zigmund A., Trigonometricheskie ryady, T. 2, Mir, M., 1965, 540 pp.

[11] Volosivets S. S., “Skhodimost ryadov Fure po multiplikativnym sistemam i $p$-fluktuatsionnyi modul nepreryvnosti”, Sib. matem. zhurn., 47:2 (2006), 241–258 | MR | Zbl

[12] Pal J., Simon P., “On a generalization of the concept of derivative”, Acta Math. Hung., 29:1–2 (1977), 155–164 | MR | Zbl

[13] Khardi G., Littlvud Dzh., Polia G., Neravenstva, In. lit., M., 1948, 456 pp.

[14] Leindler L., “Further sharpening of inequalities of Hardy and Littlewood”, Acta Sci. Math. (Szeged), 54:3–4 (1990), 285–289 | MR | Zbl

[15] Kachmazh S., Shteingauz G., Teoriya ortogonalnykh ryadov, Fizmatgiz, M., 1958, 508 pp.

[16] Leindler L., “Über Strukturbedingungen für Fourierreihen”, Math. Z., 88:5 (1965), 418–431 | DOI | MR | Zbl

[17] Bari N. K., Stechkin S. B., “Nailuchshie priblizheniya i differentsialnye svoistva dvukh sopryazhennykh funktsii”, Tr. Mosk. matem. o-va, 5, 1956, 483–522 | MR | Zbl

[18] Szasz O., “Über die Fourierschen Reihen gewisser Funktionenklassen”, Math. Ann., 100 (1928), 530–536 | DOI | MR | Zbl

[19] Szasz O., “Fourier series and mean moduli of continuity”, Trans. Amer. Math. Soc., 42:3 (1937), 366–395 | DOI | MR

[20] Hirshman I. I., “Multiplier transformations”, Duke Math. J., 26:2 (1959), 222–242

[21] Rudin W., “Some theorems on Fourier coefficients”, Proc. Amer. Math. Soc., 10:6 (1959), 855–859 | DOI | MR | Zbl