Tovkach R. On the сonditions for the convergence of Fourier series and approximation of functions from classes C(e).

The thesis is devoted to the investigation of conditions on the convergence in average of Fourier series. The work also investigated on an estimate of the deviation functions from of average linear polynomials for some classical summation methods of Fourier series. We found for the necessary and sufficient conditions in terms Taylor coefficient for the convergence in average of Taylor series of analytic functions the Hardy classes. Also obtained sufficient conditions expressed in terms of Fourier coefficients, for the convergence in average of multiple Fourier series of summable functions. A necessary and sufficient conditions to linear method of summation of Fourier series in the space of continuous functions was regular are found. Also we obtain the conditions of the coefficients under which Faber series convergence uniformly to the function inside the region.