Sharp Kolmogorov–Remez type inequalities for periodic funtions of a small smoothness

• Datum: 2020
• Hauptverfasser: Kofanov, V. A., Кофанов, В. А.
• Format: Artikel
• Sprache: Russisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2020
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/963
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.5 In the case of either $r = 2, k = 1$ or $r = 3, k = 1, 2,$ for any $q, p \geq 1,$ $\beta \in [0, 2\pi),$ and arbitrary measurable set $B \subset I_{2\pi} := [-\pi/2, 3\pi/2],$ $\mu B \le \beta,$ we prove the sharp Kolmogorov–Remez type inequality$$\|f^{(k)}\|_{q}\leq\frac{\|\varphi_{r-k}\|_{q}}{E_0(\varphi_{r})^{\alpha}_{L_p(I_{2\pi}\setminus B_{2m})}}\|f\|^{\alpha}_{L_p(I_{2\pi} \setminus B)}\big \|f^{(r)} \big \|^{1-\alpha}_{\infty}, \quad f \in L^r_\infty, $$with $\alpha = \min\{1-k/r, (r-k + 1/q)/(r + 1/p)\},$ where $\varphi_r $is the perfect Euler's spline of order~$r,$ $E_0(f)_{L_p(G)}$ is the best approximation of $f$by the constants in $L_p(G),$ $B_{2 m} = \left[\dfrac{\pi-2m}{2}, \dfrac{\pi + 2 m}{2}\right],$ and $m = m(\beta) \in [0, \pi)$ is uniquely defined by~$\beta.$ In addition, we obtain a sharp Kolmogorov–Remez type inequality in the case where the number of sign changes of derivatives is also taken into account.