Sharp Remez-type inequalities of various metrics for differentiable periodic functions, polynomials, and splines
We prove a sharp Remez-type inequality of various metrics $$\| x\| q \leq \| \varphi_r\| q \biggl\{\frac{\| x\|_{L_p([0,2\pi ]\setminus B)}}{\|\varphi r\|_{ L_p([0,2\pi ]\setminus B_1)}}\biggr\}^{\alpha } \| x(r)\|^{1 - \alpha}_{ \infty} ,\; q > p > 0, \;\alpha = (r + 1/q)/(r + 1/p...
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| Date: | 2017 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2017
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1685 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | We prove a sharp Remez-type inequality of various metrics
$$\| 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 $(\ast )$. |
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