Abstract
In the paper, we generalize the well-known criteria of Bernstein and Stechkin on the absolute convergence in terms of best approximations and moduli of smoothness of continuous functions. We give conditions for the convergence of the series of Fourier coefficients raised to the power β in terms of best approximations in the space of p-absolutely continuous functions and in terms of fractional moduli of continuity with respect to this space. We also prove the sharpness of our conditions for 0 < β ≤ 1 with no restriction and for 1 < β ≤ 2 under some restrictions.
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