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

Abstract

We prove a sharp Remez-type inequality of various metrics $$\| x\| q \leq \| \varphi_r\| q \biggl\{\frac{\| x\|_{L_p([0,2\pi ]\setminus B)}}{\|\varphi r\|_{ L_p([0,2\pi ]\setminus B_1)}}\biggr\}^{\alpha } \| x(r)\|^{1 - \alpha}_{ \infty} ,\; q > p > 0, \;\alpha = (r + 1/q)/(r + 1/p),$$ for $2\pi$ -periodic functions $x \in L^r_{\infty}$ satisfying the condition $$L(x)p \leq 2^{-\frac 1p}\| x\|_p,\quad (\ast )$$ where $$L(x)p := \mathrm{s}\mathrm{u}\mathrm{p} \Bigl\{ \| x\| L_p[a,b] : [a, b] \subset [0, 2\pi ], | x(t)| > 0, t \in (a, b)\Bigr\},$$ $B \subset [0, 2\pi ], \mu B \leq \beta /\lambda$ ($\lambda$ is chosen so that $\| x\| p = \| \varphi \lambda ,r\| L_p[0,2\pi /\lambda ] ), \varphi_r$ is the ideal Euler’s spline of order r, and $$B_1 := \biggl[\frac{-\pi - \beta /2}{2} , \frac{-\pi + \beta /2}{2} \biggr] \bigcup \biggl[ \frac{\pi - \beta /2}{2}, \frac{\pi + \beta /2}{2} \biggr].$$ As a special case, we establish sharp Remez-type inequalities of various metrics for trigonometric polynomials and polynomial splines satisfying the condition $(\ast )$.