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

Abstract

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.