N. A. Ilyasov, “Some supplements to S. B. Stechkin's inequalities in direct and inverse theorems on the approximation of continuous periodic functions”, Trudy Inst. Mat. i Mekh. UrO RAN, 26, no. 4, 2020, 155

Abstract: 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.