Inequalities of Different Metrics for Differentiable Periodic Functions
We prove the following sharp inequality of different metrics: $$\begin{array}{cc}\hfill {\left\Vert x\right\Vert}_q\le {\left\Vert {\varphi}_r\right\Vert}_q{\left(\frac{{\left\Vert x\right\Vert}_p}{{\left\Vert {\varphi}_r\right\Vert}_p}\right)}^{\frac{r+1/q}{r+1/p}}{\left\Vert {x}^{(r)}\right\Vert}_...
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| Date: | 2015 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2015
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1974 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | We prove the following sharp inequality of different metrics:
$$\begin{array}{cc}\hfill {\left\Vert x\right\Vert}_q\le {\left\Vert {\varphi}_r\right\Vert}_q{\left(\frac{{\left\Vert x\right\Vert}_p}{{\left\Vert {\varphi}_r\right\Vert}_p}\right)}^{\frac{r+1/q}{r+1/p}}{\left\Vert {x}^{(r)}\right\Vert}_{\infty}^{\frac{1/p-1/q}{r+1/p}},\hfill & \hfill q>p>0,\hfill \end{array}$$
for 2π -periodic functions $x ∈ L_{∞}^r$ satisfying the condition
$$L{(x)}_p\le {2}^{1/p}{\left\Vert x\right\Vert}_p,$$
where
$$L{(x)}_p:= \sup \left\{{\left\Vert x\right\Vert}_{L_p\left[a,b\right]}:a,b\in \left[0,2\pi \right],\kern0.5em \left|x(t)\right|>0,\kern0.5em t\in \left(a,b\right)\right\},$$
and $φ_r$ is the Euler spline of order $r$. As a special case, we establish the Nikol’skii-type sharp inequalities for polynomials and polynomial splines satisfying the condition (A). |
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