On approximation of periodic functions by Fourier sums

Abstract

Let L p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm $$ {\left\| f \right\|_p} = {\left( {\int\limits_{ - \pi }^\pi {{{\left| f \right|}^p}} } \right)^{{1 \mathord{\left/{\vphantom {1 p}} \right.} p}}} $$ , and let C = L ∞ be the space of continuous 2π-periodic functions with the norm $$ {\left\| f \right\|_\infty } = \left\| f \right\| = \mathop {\max }\limits_{e \in \mathbb{R}} \left| {f(x)} \right| $$ . Let CP be the subspace of C with a seminorm P invariant with respect to translation and such that $$ P(f) \leqslant M\left\| f \right\| $$ for every f ∈ C. By $$ \sum\limits_{k = 0}^\infty {{A_k}} (f) $$ denote the Fourier series of the function f, and let $$ \lambda = \left\{ {{\lambda_k}} \right\}_{k = 0}^\infty $$ be a sequence of real numbers for which $$ \sum\limits_{k = 0}^\infty {{\lambda_k}} {A_k}(f) $$ is the Fourier series of a certain function f λ ∈ L p . The paper considers questions related to approximating the function f λ by its Fourier sums S n (f λ) on a point set and in the spaces L p and CP. Estimates for $$ {\left\| {{f_\lambda } - {S_n}\left( {{f_\lambda }} \right)} \right\|_p} $$ and P(f λ − S n (f λ)) are obtained by using the structural characteristics (the best approximations and the moduli of continuity) of the functions f and f λ. As a rule, the essential part of deviation is estimated with the use of the structural characteristics of the function f. Bibliography: 11 titles.