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 $n\to\infty$ 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 $C^r$, the asymptotics is found along with the rate of decrease of the remainder term.