Order equalities in the spaces $L_p(\mathbb T), 1$ < $p$ < $\infty$, for best approximations and moduli of smoothness of derivatives of periodic functions with monotone Fourier coefficients

Denote by $M_p^{(r)}(\mathbb T)$ the class of all functions $f\in L_p(\mathbb T)$ whose Fourier coefficients satisfy the conditions: $a_0(f)=0$, $0<n^ra_n(f)\downarrow 0$, and $0<n^rb_n(f)\downarrow 0$ $(n\uparrow \infty)$, where $1<p<\infty$, $r\in \mathbb N$, and $\mathbb T=(-\pi,\pi]$. We establish order equalities in the class $M_p^{(r)}(\mathbb T)$ between the best approximations $E_{n-1}(f^{(r)})_p$ by trigonometric polynomials of order $n-1$ and the $k$th-order moduli of smoothness $\omega_k(f^{(r)};\pi/n)_p$ of $r$th-order derivatives $f^{(r)}$, on the one hand, and various expressions containing elements of the sequences $\{E_{\nu-1}(f^{(r)})_p\}_{\nu=1}^{\infty}$ and $\{\omega_l(f;\pi/\nu)_p\}_{\nu=1}^{\infty}$, where $l,k\in \mathbb N$ and $l>r$, on the other hand. The main results obtained in the present paper can be briefly described as follows. A necessary and sufficient condition for a function $f$ from $M_p^{(r)}(\mathbb T)$ to lie in the class $L_p^{(r)}(\mathbb T)$ (this class consists of all functions $f\in L_p(\mathbb T)$ with absolutely continuous $(r-1)$th derivatives $f^{(r-1)}$ and $f^{(r)}\in L_p(\mathbb T)$; here $f^{(0)}\equiv f$ and $L_p^{(0)}(\mathbb T)\equiv L_p(\mathbb T)$) is that one of the following equivalent conditions is satisfied: $E(f;p;r)\!:=\!\big(\sum_{n=1}^{\infty}n^{pr-1}\!E_{n-1}^{p}(f)_p\big)^{1/p}<\infty$ $\Leftrightarrow$ $\Omega(f;p;l;r)\!:=\big(\sum_{n=1}^{\infty}n^{pr-1}\omega_{l}^{p}(f;\pi/n)_p\big)^{1/p}<\infty~\Leftrightarrow$ $\sigma(f;p;r):=\big(\sum_{n=1}^{\infty}n^{pr+p-2}(a_n(f)+b_n(f))^p\big)^{1/p}<\infty$. Moreover, the following order equalities hold:
$(a)\ E(f;p;r)\asymp \|f^{(r)}\|_p \asymp \sigma(f;p;r) \asymp\Omega(f;p;l;r)$;
$(b)$ $E_{n-1}(f^{(r)})_p\asymp n^r E_{n-1}(f)_p+\big(\sum_{\nu=n+1}^{\infty}\nu^{pr-1}E_{\nu-1}^{p}(f)_p\big)^{1/p},\ n\in \mathbb N$;
$(c)$ $\omega_k(f^{(r)};\pi/n)_p\asymp n^{-k}\big(\sum_{\nu=1}^{n}\nu^{p(k+r)-1}E_{\nu-1}^{p}(f)_p\big)^{1/p}+ \big(\sum_{\nu=n+1}^{\infty}\nu^{pr-1}E_{\nu-1}^{p}(f)_p\big)^{1/p},\ n\in \mathbb N$;
$(d)$ $E_{n-1}(f^{(r)})_p+n^r\omega_l(f;\pi/n)_p\asymp \big(\sum_{\nu=n+1}^{\infty}\nu^{pr-1} \omega_l^{p}(f;\pi/\nu)_p\big)^{1/p}\asymp \asymp\omega_k(f^{(r)};\pi/n)_p+n^r\omega_l(f;\pi/n)_p,\ n\in \mathbb N,\ l<k+r$;
$(e)$ $n^{-(l-r)}\big(\sum_{\nu=1}^{n}\nu^{p(l-r)-1}E_{\nu-1}^{p}(f^{(r)})_p\big)^{1/p}\asymp \big(\sum_{\nu=n+1}^{\infty}\nu^{pr-1}\omega_l^{p}(f;\pi/\nu)_p\big)^{1/p}\asymp \asymp n^{-(l-r)}\big(\sum_{\nu=1}^{n}\nu^{p(l-r)-1}\omega_k^p (f^{(r)};\pi/\nu)_p\big)^{1/p},\ n\in \mathbb N,\ l<k+r$;
$(f)$ $\omega_k(f^{(r)};\pi/n)_p \asymp \big(\sum_{\nu=n+1}^{\infty}\nu^{pr-1}\omega_l^{p}(f;\pi/\nu)_p\big)^{1/p},\ n\in \mathbb N,\ l=k+r$;
$(g)$ $\omega_k(f^{(r)};\pi/n)_p \asymp n^{-k}\big(\sum_{\nu=1}^{n}\nu^{p(k+r)-1}\omega_l^{p}(f;\pi/\nu)_p\big)^{1/p}+ \big(\sum_{\nu=n+1}^{\infty}\nu^{pr-1}\omega_l^{p}(f;\pi/\nu)_p\big)^{1/p}$, $n\in \mathbb N$, $l>k+r$.
In the general case, one cannot drop the term $n^r\omega_l(f;\pi/n)_p$ in item $(d)$ either in the lower estimate on the left-hand side (for $l>r$) or in the upper estimate on the right-hand side (for $r<l<k+r$). However, if $\{ E_{n-1}(f)_p\}_{n=1}^{\infty}\in B_l^{(p)}$ $(\Rightarrow \{E_{n-1}(f^{(r)})_p\}_{n=1}^{\infty}\in B_{l-r}^{(p)})$ or $\{\omega_l(f;\pi/n)_p\}_{n=1}^{\infty}\in B_l^{(p)}$ $(\Rightarrow \{ \omega_k(f^{(r)};\pi/n)_p\}_{n=1}^{\infty}\in B_{l-r}^{(p)})$, where $B_l^{(p)}$ is the class of all sequences $\{\varphi_n\}_{n=1}^{\infty}$ $(0<\varphi_n\downarrow 0$ as $n\uparrow \infty$) satisfying the Bari $(B_l^{(p)})$-condition: $n^{-l}\big(\sum_{\nu=1}^n \nu^{pl-1}\varphi_{\nu}^p\big)^{1/p}=\mathcal O(\varphi_n)$, $n\in\mathbb N$, which is equivalent to the Stechkin $(S_l)$-condition, then
$$ E_{n-1}(f^{(r)})_p\asymp \bigg(\sum_{\nu=n+1}^{\infty}\nu^{pr-1}\omega_l^p\Big(f;\frac{\pi}{\nu}\Big)_p\bigg)^{1/p}\asymp \omega_k\Big(f^{(r)};\frac{\pi}{n}\Big)_p,\quad n\in \mathbb N. $$