On the equivalence of some inequalities in the theory of approximation of periodic functions in the spaces $L_p(\mathbb T),1 < p < \infty$

We propose a method for proving, in particular, the equivalence of M.F. Timan's known estimates for the $r$th-order $L_{p}$-moduli of smoothness $\omega_{r}(f;{\pi/n})_{p}$ and O.V. Besov's estimates for the $L_p$-norms $\|f^{(r)}\|_{p}$ of $r$th-order derivatives by using elements of the sequence $\{E_{n-1}(f)_{p}\}_{n=1}^{\infty}$ of the best approximations of a $2\pi$-periodic function $f\in L_{p}(\mathbb T)$ by trigonometric polynomials of order at most $n-1$, $n\in \mathbb N$, where $r\in \mathbb N$, $1 < p < \infty$, and $\mathbb T=(-\pi,\pi]$. Theorem 1.  Let $1 < p < \infty$, $\theta=\min\{2,p\}$, $r\in \mathbb N$, $f\in L_{p}(\mathbb T)$, and $\sum_{n=1}^{\infty}n^{\theta r-1} E_{n-1}^{\theta}(f)_{p} < \infty$. Then the inequality $\omega_{r}(f;\pi/n)_{p}\le C_{1}(r,p)n^{-r}\Big(\sum_{\nu=1}^{n}\nu^{\theta r-1}E_{\nu-1}^{\theta}(f)_{p}\Big)^{1/\theta}$, $n\in \mathbb N$, is satisfied if and only if $f\in L_{p}^{(r)}(\mathbb T)$ and $\|f^{(r)}\|_{p} \le C_{2}(r,p) \Big(\sum_{n=1}^{\infty}n^{\theta r-1} E_{n-1}^{\theta}(f)_{p}\Big)^{1/\theta}$, where $L_{p}^{(r)}(\mathbb T)$ is the class of functions $f\in L_{p}(\mathbb T)$ with absolutely continuous derivative of the $(r-1)$th order and $f^{(r)} \in L_{p}(\mathbb T)$. Theorem 2.  Suppose that $1 < p < \infty$, $\beta=\max\{2,p\}$, $r\in \mathbb N$, and $f\in L_{p}^{(r)}(\mathbb T)$. Then the inequality  $n^{-r}\Big(\sum_{\nu=1}^{n}\nu^{\beta r-1} E_{\nu-1}^{\beta}(f)_{p}\Big)^{1/\beta}\le C_{3}(r,p)\omega_{r}(f;\pi/n)_{p}$ is satisfied for $n\in \mathbb N$ if and only if the inequality $\Big(\sum_{n=1}^{\infty}n^{\beta r-1}E_{n-1}^{\beta}(f)_{p}\Big)^{1/\beta}\le C_{4}(r,p)\|f^{(r)}\|_{p}$ is satisfied. In view of the order identity $\sum_{\nu=1}^{n}\nu^{\alpha r-1}E_{\nu-1}^{\alpha}(f)_{p}\asymp\sum_{\nu=1}^{n}\nu^{\alpha r-1} \omega_{l}^{\alpha}(f;\pi/\nu)_{p}$, $n\in\mathbb N\cup\{+\infty\}$, where $1\le\alpha < \infty$, $l\in\mathbb N$, and $l>r$, the assertions of Theorems 1 and 2 remain valid if we replace the sequence $\{E_{n-1}(f)_{p}\}_{n=1}^{\infty}$ by the sequence $\{\omega_{l}(f;\pi/n)_{p}\}_{n=1}^{\infty}$ (Theorems 3 and 4). The method used in the proof of Theorems 1 and 2 can be applied to derive equivalent upper estimates and equivalent lower estimates for the values $E_{n-1}(f^{(r)})_{p}$ and $\omega_{k}(f^{(r)};\pi/n)_{p}$, $n\in \mathbb N$, by means of elements of the sequence $\{E_{n-1}(f)_{p}\}_{n=1}^{\infty}$, where $k,r\in \mathbb N$ and $1 < p < \infty$.