Sharp Remez-type inequalities of various metrics in the classes of functions with а given comparison function

Abstract

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.