On linear methods for the summability of Fourier series that are similar to the Bernshtein–Rogosinski method

We consider sequences of linear operators, the construction of which recalls the well-known Bernshtein–Rogosinski sums and constitutes a far-reaching generalization of them. We study conditions for the convergence of sequences of such operators, using results that we first obtain on linear summability methods for Fourier–Lebesgue series and their conjugate series.