Multiple modules of continuity and the best approximations of periodic functions in metric spaces
It is proved that, under the condition $M_{\Psi} \Bigl( \frac 12\Bigr) < 1$, where $M_{\Psi}$ is a stretching function $\Psi$ in the space $L_{\Psi}$ , the Jackson inequalities $$\sup_n \sup_{f\in L_{\Psi}, f\not = \text{const}} \frac{E_{n-1}(f)_{\Psi} }{\omega_k \Bigl(f, \frac{\pi}n \Bigr...
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| Date: | 2018 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2018
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1588 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | It is proved that, under the condition $M_{\Psi} \Bigl( \frac 12\Bigr)
< 1$, where $M_{\Psi}$ is a stretching function $\Psi$ in the space $L_{\Psi}$ , the Jackson
inequalities
$$\sup_n \sup_{f\in L_{\Psi}, f\not = \text{const}} \frac{E_{n-1}(f)_{\Psi} }{\omega_k \Bigl(f, \frac{\pi}n
\Bigr)_{\Psi}} < \infty,$$
are true; here, $E_{n-1}(f)_{\Psi}$ is the best approximation of $f$ by trigonometric polynomials of degree at most $n - 1$ and $\omega_k \Bigl(f, \frac{\pi}n \Bigr)_{\Psi}$ is the modulus of continuity of $f$ of order $k$, $k \in N$. We study necessary and sufficient conditions for the
function $f$ under which the following relation is true: $E_{n-1}(f)_{\Psi} \asymp \omega_k
\Bigl(f, \frac{\pi}n
\Bigr)_{\Psi}.$ |
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