| Abstract: | Let L p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm || f ||p = ( ò - pp | f |p )1 mathord | / | vphantom 1 p p {left| f right|_p} = {left( {intlimits_{ - 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 || f ||￥ = || f || = maxe ? mathbbR | f(x) | {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|| f || P(f) leqslant Mleft| f right| for every f ∈ C. By ?k = 0￥ Ak (f) sumlimits_{k = 0}^infty {{A_k}} (f) denote the Fourier series of the function f, and let l = { lk }k = 0￥ lambda = left{ {{lambda_k}} right}_{k = 0}^infty be a sequence of real numbers for which ?k = 0￥ lk Ak(f) sumlimits_{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 || fl - Sn( fl ) ||p {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. |
