Methods of summation and best approximation

Abstract

Let λ={λ k n } be a triangular method of summation,f e Lp (1 ≤ p ≤ ∞), $$U_n (f,x,\lambda ) = \frac{{a_0 }}{2} + \sum _{k = 1}^n \lambda _k^n (a_k \cos kx + b_k \sin kx).$$ Consideration is given to the problem of estimating the deviations ∥f − Un (f, λ) ∥ Lp in terms of a best approximation En (f) Lp in abstract form (for a sequence of projectors in a Banach space). Various generalizations of known inequalities are obtained.