Pelagenko O. Estimates of trigonometric series and their application to the problems of approximation theory

For the special ("polyhedral") multiple trigonometric series with polyhedral sums the sufficient conditions of Lebesgue intergrability of their sums are found. Integral estimates of the modulus of the functions adjusted with these multiple trigonometric series are established. For the special ("triangular") multiple Fourier and Taylor series with triangular sums the necessary conditions of the mean convergence are found. The necessary and sufficient conditions of the mean convergence of some classes of multiple Fourier series are set. It is distinguished some class of linear summation methods of one-dimensional real Fourier series for which the necessary and sufficient conditions of their regularity are found in space of continuous periodic functions without the calculation of Lebesgue constants.