Sharp Remez-type inequalities for differentiable periodic functions, polynomials and splines

  • V. A. Kofanov

Abstract

For any $\omega > 0,\; \beta \in (0, 2\omega)$, and any measurable set $B \in I_d := [0, d],\; \mu B = \beta$, we obtain the following sharp inequality of the Remez type: $$||x||_{\infty} \leq \frac{3||\varphi||_{\infty} - \varphi \biggl(\frac{\omega - \beta}2 \biggr)}{||\varphi||_{\infty} + \varphi \biggl(\frac{\omega - \beta}2 \biggr)} ||x||_{L_{\infty}(I_d\setminus B)}$$ on the set $S_{\varphi} (\omega )$ of functions $x$ with minimal period $d (d \geq 2\omega)$ and a given sine-shaped $2\omega$ -periodic comparison function $\varphi$. In particular, we prove the sharp Remez-type inequalities on the Sobolev spaces of differentiable periodic functions. We also obtain inequalities of the indicated type on the spaces of trigonometric polynomials and polynomial splines.
Published
25.02.2016
How to Cite
KofanovV. A. “Sharp Remez-Type Inequalities for Differentiable Periodic Functions, Polynomials and Splines”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, no. 2, Feb. 2016, pp. 227-40, http://umj.imath.kiev.ua/index.php/umj/article/view/1836.
Section
Research articles