Approximating periodic functions in Hölder type metrics by singular integrals with positive Kernels

Abstract

Let M be either the space of 2π-periodic functions L_p, where 1 ≤ p < ∞, or C; let ω_r(f, h) be the continuity modulus of order r of the function f, and let

$$D_{n,r,l} (f,x) = \frac{{( - 1)^{\tfrac{r}{2} + 1} 2}}{{C_r^{r/2} }}\int\limits_\mathbb{R} {\left\{ {\sum\limits_{k = 1}^{r/2} {( - 1)^{k + r/2} C_r^{k + r/2} f(x + kt)} } \right\}} V_{n,2l} (t) dt$$

, where

$$V_{n,2l} (t) = \frac{{(2l - 1)!2^{2l - 1} }}{{\lambda _{2l} \pi (n + 1)^{2l - 1} }}\left( {\frac{{\sin \tfrac{{(n + 1)t}}{2}}}{t}} \right)^{2l} , \lambda _{2l} = \sum\limits_{k = 0}^{l - 1} {( - 1)^k C_{2l}^k (l - k)^{2l - 1} } $$

, be the generalized Jackson-Vallée-Poussin integral. Denote

$$K_m (f) = K_{m,\varphi } (f) = \mathop {\sup }\limits_{0 < v < \infty } \frac{{\omega _m (f,v)}}{{\varphi (v)}}$$

. The paper studies the quantity K_m(f − D_n,r,l(f)). The general results obtained are applicable to other approximation methods. Bibliography: 11 titles.