General Fourier coefficients and problems of summability almost everywhere

S. Banach proved that for any $L_2$ 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 system contains a subsystem for which the Fourier series of functions of bounded variation are Cesàro summable a.e. These results generalize those of L. Gogoladze and V. Tsagareishvili [Studia Sci. Math. Hungar. 52 (2015), 511–536].