We study the problem of order optimality of an upper bound for the best approximation in~$L_{q}(\mathbb T)$ in terms of the $l$th-order modulus of smoothness (the modulus of continuity for $l=1$) in $$L_{p}(\mathbb T)\colon E_{n-1}(f)_{q}\le C(l,p,q)\big(\textstyle\sum\limits_{\nu=n+1}^{\infty}\nu^{q\sigma-1}\omega_{l}^{q}(f;\pi/\nu)_{p}\big)^{1/q}, n\in\mathbb N,$$ on the class $M_{p}(\mathbb T)$ of all functions $f\in L_{p}(\mathbb T)$ whose Fourier coefficients satisfy the conditions $$a_{0}(f)=0, a_{n}(f)\downarrow 0, \text {and} b_{n} (f)\downarrow 0 (n\uparrow \infty), l\in\mathbb N, 1< p < q < \infty, l>\sigma=1/p-1/q, \text{and} \mathbb T=(-\pi,\pi].$$ For $l=1$ and $p\ge 1$, the bound was first established by P. L. Ul'yanov in the proof of the inequality of different metrics for moduli of continuity; for $l>1$ and $p\ge 1$, the proof of the bound remains valid in view of the $L_{p}$-analog of the Jackson-Stechkin inequality. Below we formulate the main results of the paper. A function $f\in M_{p}(\mathbb T)$ belongs to $L_{q}(\mathbb T)$, where $1 < p < q < \infty$, if and only if $\sum_{n=1}^{\infty}n^{q\sigma-1}\omega_{l}^{q}(f;\pi/n)_{p}< \infty$, and the following order inequalities hold:
(a) $E_{n-1}(f)_{q}+n^{\sigma}\omega_{l}(f;\pi/n)_{p}\asymp\big(\sum\limits_{\nu=n+1}^{\infty}\nu^{q\sigma-1}\omega_{l}^{q} (f;\pi/\nu)_{p}\big)^{1/q}$, $n\in\mathbb N$;
(b) $n^{-(l-\sigma)}\big(\sum_{\nu=1}^{n}\nu^{p(l-\sigma)-1}E_{\nu-1}^{p}(f)_{q}\big)^{1/p}\asymp \big(\sum\limits_{\nu=n+1}^{\infty}\nu^{q\sigma-1}\omega_{l}^{q}(f;\pi/\nu)_{p}\big)^{1/q}$, $n\in\mathbb N$.
In the lower bound in inequality (a), the second term $n^{\sigma}\omega_{l}(f;\pi/n)_{p}$ generally cannot be omitted. However, if the sequence $\{\omega_{l}(f;\pi/n)_p\}_{n=1}^{\infty}$ or the sequence $\{E_{n-1}(f)_{p}\}_{n=1}^{\infty}$ satisfies Bari's $(B_{l}^{(p)})$-condition, which is equivalent to Stechkin's $(S_{l})$-condition, then $$E_{n-1}(f)_{q}\asymp\bigg(\sum_{\nu=n+1}^{\infty}\nu^{q\sigma-1}\omega_{l}^{q}(f;\pi/\nu)_{p}\bigg)^{1/q}, n\in\mathbb N.$$ The upper bound in inequality~(b), which holds for any function $f\in L_{p}(\mathbb T)$ if the series converges, is a strengthened version of the direct theorem. The order inequality $(b)$ shows that the strengthened version is order-exact on the whole class~$M_{p}(\mathbb T)$.
Keywords: best approximation, modulus of smoothness, direct theorem in different metrics, trigonometric Fourier series with monotone coefficients, order-exact inequality on a class.
The paper was received by the Editorial Office on March 15, 2017.
Niyazi Aladdin ogly Il'yasov, Cand. Sci. (Phys.-Math.), Baku State University, Baku, Azerbaijan, e-mail: niyazi.ilyasov@gmail.com .
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