Specifications of direct and inverse approximation theorems for \(p\)-absolutely continuous functions (Q1406432)

Let \(f(x)\) be a real bounded \(2\pi\)-periodic function and \( \xi = \{x_0<x_1<\dots<x_n=x_0+2\pi\} \) be the periodic division. Designate \[ \begin{gathered} \text{æ}^p_\xi(f) = \Bigg(\sum\limits_{i=1}^n |f(x_i) - f(x_{i-1})|^p\Bigg)^{1/p},\\ \omega_{1-1/p}(f,\delta) = \sup\{\text{æ}^p_\xi(f) : \max_i(x_i-x_{i-1}) \leq \delta\},\\ \omega_{k-1/p}(f,\delta) = \sup\{\omega_{1-1/p}(\Delta_h^{k-1}f(x),|h|) : |h|\leq \delta\},\quad\text{for}\quad k\geq 2. \end{gathered} \] In the paper the author refines the inequalities \[ \begin{gathered} E_n(f)_p\leq C\omega_{k-1/p}(f,1/n),\\ \omega_{k-1/p}(f,1/n)\leq Cn^{-k+1/p} \sum\limits_{i=1}^n i^{k-1/p-1} E_i(f)p \end{gathered} \] established by \textit{A. P. Terekhin} [Izv. Vyssh. Uchebn. Zaved., Mat. 1965, No. 2(45), 171-187 (1965; Zbl 0156.06802)].