Sharp inequalities of Jackson-Stechkin type and widths of classes of functions in $L_{2}$

Abstract

Some problems of the approximation theory require estimating the best approximation of $2\pi$-periodic functions by trigonometric polynomials in the space $L_2$, and while doing this, instead of the usual modulus of continuity $\omega_{m}(f, t)$, sometimes it is more convenient to use an equivalent characteristic $\Omega_{m}(f, t)$ called the generalized modulus of continuity. Similar averaged characteristic of the smoothness of a function was considered by K.V. Runovskiy and E.A. Storozhenko, V.G. Krotov and P. Oswald while studying important issues of constructive function theory in metric space $L_{p}$, $0 < p < 1$. In the space $L_2$, in finding exact constants in the Jackson-type inequality, it was used by S.B. Vakarchuk. We continue studies of problems approximation theory and consider new sharp inequalities of the type Jackson–Stechkin relating the best approximations of differentiable periodic functions by trigonometric polynomials with integrals containing generalized modules of continuity. For classes of functions defined by means of these characteristics, we calculate exact values of some known $n$-widths are calculated.