Abstract:
Let $\lambda=\{\lambda_k^n\}$ be a triangular method of summation, $f\in L_p$$(1\le p\le\infty)$,
$$
U_n(f,x,\lambda)=\frac{a_0}2+\sum_{k=1}^n\lambda_k^n(a_k\cos kx+b_k\sin kx).
$$
Consideration is given to the problem of estimating the deviations $\|f-U_n(f,\lambda)\|_{L_p}$ in terms of a¨best approximation $E_n(f)_{L_p}$ in abstract form (for a sequence of projectors in a Banach space). Various generalizations of known inequalities are obtained.
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