N.A. Il’yasov. 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 ... P. 103-120

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
$$
E_{n-1}(f^{(r)})_p\asymp \bigg(\sum_{\nu=n+1}^{\infty}\nu^{pr-1}\omega_l^p\Big(f;\frac{\pi}{\nu}\Big)_p\bigg)^{1/p}\asymp \omega_k\Big(f^{(r)};\frac{\pi}{n}\Big)_p,\quad n\in \mathbb N.
$$

Keywords: best approximation, modulus of smoothness, direct and inverse theorems with derivatives of the theory of approximation of periodic functions, trigonometric Fourier series with monotone coefficients, order equalities

Received September 8, 2022

Revised October 17, 2022

Accepted October 24, 2022

Funding Agency: This research was supported by the Ministry of Science and Higher Education of the Russian Federation within a program of the Moscow Center for Fundamental and Applied Mathematics (agreement no. 075-15-2022-284).

Niyazi Aladdin ogly Il’yasov, Cand. Sci. (Phys.-Math.), Senior Research Fellow, Department of Theory of Functions, Institute of Mathematics and Mechanics of the National Academy of Sciences of Azerbaijan., Baku, Azerbaijan; Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia. e-mail: niyazi.ilyasov@gmail.com

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Cite this article as: N.A. Il’yasov. 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,Trudy Instituta Matematiki i Mekhaniki URO RAN, 2022, vol. 28, no. 4, pp. 103–120