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S. Banach proved that for any L2 function, there exists an orthonormal system such that the Fourier series of this function is not Cesàro summable a.e. In this paper, we present sufficient conditions that must be satisfied by functions of an orthonormal system so that the Fourier coefficients of any function of bounded variation satisfy the conditions of the Menshov-Kaczmarz theorem. The results obtained are the best possible in a certain sense. We also prove that any orthonormal systemdoi:10.4064/ap200731-23-10 fatcat:vjiogb7z2zfczjtghpogeysfl4