Some supplements to S. B. Stechkin's inequalities in direct and inverse theorems on the approximation of continuous periodic functions

N. A. Ilyasov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 4, pp. 155-181

Résumé

We give some supplements and comments to inequalities between elements of the sequence of best approximations $\{E_{n-1}(f)\}_{n=1}^{\infty}$ and the $k$th-order moduli of smoothness $\omega_k(f^{(r)};\delta),$ $\delta\in [0,+\infty)$, of a function $f\in C^r(\mathbb{T})$, where $k\in \mathbb{N},$ $r\in \mathbb{Z}_+$, $f^{(0)}\equiv f,$ $C^0(\mathbb{T})\equiv C(\mathbb{T}),$ and $\mathbb{T}=(-\pi,\pi]$, which were published by S. B. Stechkin in 1951 in the study of direct and inverse theorems of approximation of $2\pi$-periodic continuous functions. In particular, we prove the following results: $\mathrm{(a)}$ the direct theorem or the Jackson–Stechkin inequality: $E_{n-1}(f)\le C_1(k)\omega_k(f;\pi/n)$, $n\in \mathbb{N}$, can be strengthened as $E_{n-1}(f)\le \rho_{n}^{(k)}(f)\equiv n^{-k}\max\{\nu^k E_{\nu-1}(f)\colon 1\le \nu\le n\}\le 2^kC_1(k)\omega_k(f;\pi/n),\ n\in \mathbb{N}$. This inequality is order-sharp on the class of all functions $f\in C(\mathbb{T})$ with a given majorant or with a given decrease order of the modulus of smoothness $\omega_k(f;\delta)$; namely: for any $k\in \mathbb{N}$ and $\omega\in \Omega_k(0,\pi]$, there exists a function $f_0(\,{\cdot}\,;\omega)\in C(\mathbb{T})$ ($f_0$ is even for odd $k$ and is odd for even $k$) such that $\omega_k(f_0;\delta)\asymp C_2(k)\omega(\delta)$, $\delta\in (0,\pi]$. Moreover, order equalities hold: $E_{n-1}(f_0)\asymp C_3(k)\rho_n^{(k)}(f_0)\asymp C_4(k)\omega_k(f_0;\pi/n)\asymp C_5(k)\omega(\pi/n),\ n\in \mathbb{N}$, where $\Omega_k(0,\pi]$ is the class of functions $\omega=\omega(\delta)$ defined on $(0,\pi]$ and such that $0\omega(\delta)\!\downarrow\!0$ $(\delta\downarrow\!0)$ and $\delta^{-k}\omega(\delta)\!\downarrow$ $(\delta \uparrow)$; $\mathrm{(b)}$ a necessary and sufficient condition under which the inverse theorem (without the derivatives), or the Salem–Stechkin inequality $\omega_k(f;\pi/n)\le C_6(k)n^{-k}\sum_{\nu=1}^n\nu^{k-1}E_{\nu-1}(f)$, $n\in \mathbb{N}$, holds is Stechkin's inequality $\|T_n^{(k)}(f)\|\le C_7(k) \sum_{\nu=1}^{n}\nu^{k-1}E_{\nu-1}(f),\ n\in \mathbb{N}$, where $T_n(f)\equiv T_n(f;x)$ is a trigonometric polynomial of best $C(\mathbb{T})$-approximation to the function $f$ (i.e., $\|f-T_n(f)\|=E_n(f),\ n\in \mathbb{Z}_+$); $\mathrm{(c)}$ the inverse theorem (with the derivatives), or the Vallée-Poussin–Stechkin inequality $\omega_k(f^{(r)};$ $\pi/n)\le C_8(k,r)\big\{ n^{-k}\sum_{\nu=1}^{n}\nu^{k+r-1}E_{\nu-1}(f)+\sum_{\nu=n+1}^{\infty}\nu^{r-1}E_{\nu-1}(f)\big\}$ for any $n\in \mathbb{N}$, as well as Stechkin's earlier inequality $E_{n-1}(f^{(r)})\le C_9(r)\big\{ n^r E_{n-1}(f)+\sum_{\nu=n+1}^{\infty}\nu^{r-1}E_{\nu-1}(f)\big\},\ n\in \mathbb{N}$, where $E(f;r)\equiv$ $ \sum_{n=1}^{\infty}n^{r-1}E_{n-1}(f)\infty$ (by S. N. Bernstein's theorem, this inequality guarantees that $f$ lies in $C^r(\mathbb{T})$, where $r\in\mathbb{N}$) can be supplemented with the following key inequalities: $\|f^{(r)}\|\le C_{10}(r)E(f;r)$ and $\|T_n^{(r)}(f)\|\le C_{7}(r)\sum_{\nu=1}^n\nu^{r-1}E_{\nu-1}(f)$, $n\in\mathbb{N}$. Moreover, all the inequalities formulated in this paragraph are pairwise equivalent; i.e., any of these inequalities implies any other and, hence, all the inequalities.

Détail

Keywords: best approximation, modulus of smoothness, direct theorem, inverse theorem, order equality, equivalent inequalities, order-sharp inequality on a class.

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