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 Trudy Inst. Mat. i Mekh. UrO RAN, 2018, Volume 24, Number 4, Pages 176–188 (Mi timm1584)

On the equivalence of some relations in different metrics between norms, best approximations, and moduli of smoothness of periodic functions and their derivatives

N. A. Il'yasov

Baku State University

Abstract: We propose a method capable, in particular, of establishing the equivalence of known upper estimates for the $L_q(\mathbb T)$-norm $\|f^{(r)}\|_q$, the best approximation $E_{n-1}(f^{(r)})_q$, and the $k$th-order modulus of smoothness $\omega_k(f^{(r)};\pi/n)_q$ in terms of elements of the sequence $\{E_{n-1}(f)_p\}_{n=1}^\infty$ of 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 $r\in \mathbb Z_+$ ($f^{(0)}=f)$, $1 < p < q < \infty$, and $\mathbb T=(-\pi,\pi]$. The principal result of the paper is the following statement. Let $1 < p < q < \infty$, $r\in \mathbb Z_+$, $k\in \mathbb N$, $\sigma=r+1/p-1/q$, $f\in L_p(\mathbb T)$, and $E(f;p;\sigma;q)\equiv(\sum_{\nu=1}^{\infty}\nu^{q\sigma-1}E_{\nu-1}^{q}(f)_{p})^{1/q} < \infty$. Then the following inequalities are equivalent in the sense that each of them implies the other two: (a) $\|f^{(r)}\|_q\le C_1(r,p,q)\{(1-\chi (r))\|f\|_p+E(f;p;\sigma;q)\}$; (b) $E_{n-1}(f^{(r)})_q\le C_2(r,p,q)\{n^\sigma E_{n-1}(f)_p +(\sum\nolimits_{\nu =n+1}^\infty \nu ^{q\sigma -1}E_{\nu -1}^q (f)_p)^{1/q}\}$, $n\in\mathbb{N}$; (c) $\omega _k (f^{(r)};\pi/n)_q \le C_3 (k,r,p,q)\{(\sum\nolimits_{\nu =n+1}^\infty \nu^{q\sigma -1}E_{\nu -1}^q (f)_p)^{1/q}+n^{-k}(\sum\nolimits_{\nu =1}^n \nu ^{q(k+\sigma )-1}E_{\nu -1}^q (f)_p )^{1/q}\}$, $n\in \mathbb{N}$. \noindent Inequalities (a), (b), and (c) depend on the key estimate
$$\| S_m^{(l)} (f;\cdot )\|_q \le C_4(l,p,q)\{(1-\chi (l))\|f\|_p +(\sum\nolimits_{\nu =1}^m \nu ^{q\lambda -1} E_{\nu -1}^q (f)_p )^{1/q}\}, m\in \mathbb{N},$$
where $S_m (f;x)$ is the partial sum of order $m\in \mathbb{N}$ of the Fourier series of a function $f\in L_p(\mathbb T)$, $l\in \mathbb Z_+$, $\lambda =l+ 1/p-1/q$, $\chi (t)=0$ for $t\le 0$, and $\chi (t)=1$ for $t>0$, $t\in \mathbb{R}$. The latter estimate in the case $l=r$ and $\lambda =\sigma$ provides a necessary and sufficient condition for the fulfillment of inequality (a) under the condition $E(f;p;\sigma ;q) < \infty$, which guarantees that $f\in L_q^{(r)}(\mathbb T)$, where $L_q^{(r)} (\mathbb T)$ is the class of functions $f\in L_q (\mathbb T)$ with absolutely continuous $(r-1)$-th derivative and $f^{(r)}\in L_q (\mathbb T)$. Necessary and sufficient conditions for the validity of inequalities (b) and (c) are also provided in terms of the behavior of elements of the sequence $\{\|S_m^{(l)} (f;\cdot )\|_q\}_{m=1}^\infty$.

Keywords: best approximation, modulus of smoothness, inequalities in different metrics, equivalent inequalities.

DOI: https://doi.org/10.21538/0134-4889-2018-24-4-176-188

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Bibliographic databases:

UDC: 517.518.832
MSC: 42A10, 41A17, 41A25, 41A27
Revised: 13.11.2018
Accepted:19.11.2018

Citation: N. A. Il'yasov, “On the equivalence of some relations in different metrics between norms, best approximations, and moduli of smoothness of periodic functions and their derivatives”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 4, 2018, 176–188

Citation in format AMSBIB
\Bibitem{Ily18} \by N.~A.~Il'yasov \paper On the equivalence of some relations in different metrics between norms, best approximations, and moduli of smoothness of periodic functions and their derivatives \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2018 \vol 24 \issue 4 \pages 176--188 \mathnet{http://mi.mathnet.ru/timm1584} \crossref{https://doi.org/10.21538/0134-4889-2018-24-4-176-188} \elib{http://elibrary.ru/item.asp?id=36517708}