Sharp Remez-type inequalities of various metrics in the classes of functions with а given comparison function
For any $p \in [1,\infty ],\; \omega > 0, \;\beta \in (0, 2\omega )$, and any measurable set $B \subset I_d := [0, d], \mu B \leq \beta$, we obtain the following sharp Remez-type inequality of various metrics $$E_0(x)\infty \leq \frac{\| \varphi \|_{\infty} }{E_0 (\varphi )L_p(I_{2\ome...
Saved in:
| Date: | 2017 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | Russian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2017
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1796 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| Summary: | For any $p \in [1,\infty ],\; \omega > 0, \;\beta \in (0, 2\omega )$, and any measurable set $B \subset I_d := [0, d], \mu B \leq \beta$, we obtain the following sharp Remez-type inequality of various metrics
$$E_0(x)\infty \leq \frac{\| \varphi \|_{\infty} }{E_0 (\varphi )L_p(I_{2\omega} \setminus B_1)}\| x\|_{ L_p(I_d\setminus B)}$$
on the classes $S_{\varphi} (\omega )$ of $d$-periodic $(d \geq 2\omega)$ functions $x$ with a given sine-shaped $2\omega$ -periodic comparison function $\varphi$,
where $B_1 := [(\omega \beta )/2, (\omega + \beta )/2], E_0(f)L_p(G)$ is the best approximation of the function $f$ by constants in the
metric of the space $L_p(G)$. In particular, we prove sharp Remez-type inequalities of various metrics in the Sobolev spaces
of differentiable periodic functions. We also obtain inequalities of this type in the spaces of trigonometric polynomials and
splines. |
|---|