Sharp Remez type inequalities of various metrics with non-symmetric restrictions on functions

• Date: 2020
• Main Authors: Kofanov, V. A., Popovich, I. V., Кофанов, В. А., Попович, И. В
• Format: Article
• Language: Russian
• Published: Institute of Mathematics, NAS of Ukraine 2020
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/2352
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.5 For any $p\in (0, \infty],$ $\omega > 0,$ $\beta \in (0, 2 \omega)$, and arbitrary measurable set $B \subset I_d := [0, d],$ $\mu B \le \beta,$ we obtain the sharp inequality of Remez type$$\|x_{\pm}\|_\infty \le\frac{\|(\varphi+c)_{\pm}\|_\infty}{\|\varphi+c\|_{L_p(I_{2\omega}\setminus B^c_y)}} \left\|x \right\|_{L_{p} \left(I_d \setminus B\right)}$$on the set $S_{\varphi}(\omega)$ of $d$-periodic functions $x$ having zeros with given the sine-shaped $2\omega$-periodic comparison function $\varphi$, where $c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty]$ satisfies the condition$$\|x_{+}\|_\infty \cdot\|x_{-}\|^{-1}_\infty = \|(\varphi+c)_{+}\|_\infty \cdot\|(\varphi+c)_{-}\|^{-1}_\infty ,$$$B^c_y:=\{t\in [0, 2\omega]:|\varphi(t)+c| > y \}$ and $y$ is such that $\mu B^c_y = \beta$. In particular, we obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and splines with given quotient $\|x_{+}\|_\infty / \|x_-\|_\infty$.