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  • 1 I. Javakhishvili Tbilisi State University,, 2 University St., Tbilisi 0186, Georgia
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S. Banach in [1] proved that for any function fL2(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 (φn(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.

  • [1]

    Banach, S., Sur la divergence des series orthogonales, Studia Math., 9 (1940), 139155.

  • [2]

    Aleksic, G., Convergence problems of orthogonal series, (Russian) Izdat. Inostran. Lit., Moscow, 1963.

  • [3]

    Tsagareishvili, V., Absolute convergence of Fourier seriws of functions of class Lip 1 and functions of bounded variation, Izv. Mat., 76:2 (2012), 419429.

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  • [4]

    Ul’janov, P. L., On Haar series, (Russian) Mat. Sb. (N.S.), 63(105) (1964), 356391.

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