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 $1<p<\infty$.
Keywords: best approximation, modulus of smoothness, inequalities of approximation theory, equivalent inequalities, Timan's inequalities, Besov's inequalities.
The paper was received by the Editorial Office on March 13, 2018.
N.A. Il’yasov, Cand. Sci. (Phys.-Math.), Baku State University, Baku, Azerbaijan,
e-mail: niyazi.ilyasov@gmail.com.
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