Rogosinsky-Bernstein Polynomial Method of Summation of Trigonometric Fourier Series

General Rogosinsky-Bernstein linear polynomial means R-n(f) of Fourier series are introduced and three convergence criteria as n -> infinity are obtained: for convergence in the space C of continuous periodic functions and for convergence almost everywhere with two guaranteed sets (Lebesgue points and d-points). For smooth functions, the rate of convergence in norm of R-n(f), as well as of their interpolation analogues, is also studied. For approximation of functions in C-T, the asymptotics is found along with the rate of decrease of the remainder term.