On approximation of periodic functions by Fourier sums

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.