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-3577(01)80004-5 . [2] Berens , H. , Xu , Y. 1996 Fejér means for multivariate Fourier series Math. Z. 221 449 – 465 . [3] Carleson , L. 1966 On

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. 128 369 – 380 10.1007/s10474-010-9217-4 . [2] Tikhonov , S. 2008 On L 1 -convergence of Fourier series J. Math. Anal. Appl

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] Dezern , David H. , Fourier series on Vilenkin groups , Ph.D. Dissertation, Syracuse University, New York, 1988 . [7] Dezern , D. H. , Waterman , D. 1992 On the Lebesgue test for the

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S. Banach in [1] proved that for any function fL 2(0, 1), f ≁ 0, there exists an ONS (orthonormal system) such that the Fourier series of this function is not summable a.e. by the method (C, α), α > 0.

D. Menshov found the conditions which should be satisfied by the Fourier coefficients of the function for the summability a.e. of its Fourier series by the method (C, α), α > 0.

In this paper the necessary and sufficient conditions are found which should be satisfied by the ONS functions (φ n(x)) so that the Fourier coefficients (by this system) of functions from class Lip 1 or A (absolutely continuous) satisfy the conditions of D. Menshov.

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Abstract  

We study the uniform convergence of Walsh-Fourier series of functions on the generalized Wiener class BV (p(n)↑∞)

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1981 Blahota, I., Gát, G. and Goginava, U. , maximal operators of Fejér means of double Vilenkin-Fourier series, Colloq. Math. , 107

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Leindler, L. , Strong Approximation by Fourier Series , Akadémiai Kiadó (Budapest, 1985). MR 87g :42006 Leindler L

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References [1] Aljančić , S. 1966 On the integral moduli of continuity in L p (1< p <∞) of Fourier series with monotone coefficients

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Gogoladze, L. , On the exponential uniform strong summability of multiple trigonometric Fourier series, Georgian Math. J. , 16 (2009), 517–532. MR 2572672 ( 2010k :42016) Gogoladze L

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R. E. Edwards, Fourier series a modern introduction, vol. 1 , Springer-Verlang, New-York, Heidelberg, Berlin 1982. Edwards R. E

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