Methods of summation and best approximation

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 aЁbest 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.