The direct theorem of the theory of approximation of periodic functions with monotone Fourier coefficients in different metrics

N. A. Il'yasov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 3, pp. 144-158

Résumé

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 $$L_{p}(\mathbb{T})\colon E_{n-1}(f)_{q}\le C(l,p,q)\big(\textstyle\sum\limits_{\nu=n+1}^{\infty}\nu^{q\sigma-1}\omega_{l}^{q}(f;\pi/\nu)_{p}\big)^{1/q},\ \ n\in\mathbb{b}N,$$ 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),\ \text{where}\ l\in\mathbb{N},\ 1

\infty,\ l>\sigma=1/p-1/q,\ \text{and}\ \mathbb{T}=(-\pi,\pi].$$ For $l=1$ and $p\ge 1$, the bound was first established by P. L. Ul'yanov in the proof of the inequality of different metrics for moduli of continuity; for $l>1$ and $p\ge 1$, the proof of the bound remains valid in view of the $L_{p}$-analog of the Jackson–Stechkin inequality. Below we formulate the main results of the paper. A function $f\in M_{p}(\mathbb{T})$ belongs to $L_{q}(\mathbb{T})$, where $1$, if and only if $\sum_{n=1}^{\infty}n^{q\sigma-1}\omega_{l}^{q}(f;\pi/n)_{p}\infty$, and the following order inequalities hold: (a) $E_{n-1}(f)_{q}+n^{\sigma}\omega_{l}(f;\pi/n)_{p}\asymp\big(\sum\limits_{\nu=n+1}^{\infty}\nu^{q\sigma-1}\omega_{l}^{q} (f;\pi/\nu)_{p}\big)^{1/q}$, $n\in\mathbb{N}$; (b) $n^{-(l-\sigma)}\big(\sum_{\nu=1}^{n}\nu^{p(l-\sigma)-1}E_{\nu-1}^{p}(f)_{q}\big)^{1/p}\asymp \big(\sum\limits_{\nu=n+1}^{\infty}\nu^{q\sigma-1}\omega_{l}^{q}(f;\pi/\nu)_{p}\big)^{1/q}$, $n\in\mathbb{N}$. \noindent In the lower bound in inequality (a), the second term $n^{\sigma}\omega_{l}(f;\pi/n)_{p}$ generally cannot be omitted. However, if the sequence $\{\omega_{l}(f;\pi/n)_p\}_{n=1}^{\infty}$ or the sequence $\{E_{n-1}(f)_{p}\}_{n=1}^{\infty}$ satisfies Bari's $(B_{l}^{(p)})$-condition, which is equivalent to Stechkin's $(S_{l})$-condition, then $$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 $M_{p}(\mathbb{T})$.

Détail

Keywords: best approximation, modulus of smoothness, direct theorem in different metrics, trigonometric Fourier series with monotone coefficients, order-exact inequality on a class.

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