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

  • A. E. Gaydabura
  • V. A. Kofanov

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.
Published
25.11.2017
How to Cite
GaydaburaA. E., and KofanovV. A. “Sharp Remez-Type Inequalities of Various Metrics in the classes of Functions With а Given Comparison Function”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, no. 11, Nov. 2017, pp. 1472-85, http://umj.imath.kiev.ua/index.php/umj/article/view/1796.
Section
Research articles