Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function
For arbitrary $[\alpha, \beta] \subset \textbf{R}$ and $p > 0$, we solve the extremal problem $$\int_{\alpha}^{\beta}|x^{(k)}(t)|^q dt \rightarrow \sup, \quad q \geq p, \quad k = 0, \quad \text{or} \quad q \geq 1, \quad k \geq 1,$$ on the set of functions $S^k_{\varphi}$ such that$\varphi...
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| Date: | 2011 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2011
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2778 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | For arbitrary $[\alpha, \beta] \subset \textbf{R}$ and $p > 0$, we solve the extremal problem
$$\int_{\alpha}^{\beta}|x^{(k)}(t)|^q dt \rightarrow \sup, \quad q \geq p, \quad k = 0, \quad \text{or} \quad q \geq 1, \quad k \geq 1,$$
on the set of functions $S^k_{\varphi}$ such that$\varphi ^{(i)}$ is the comparison function for $x^{(i)},\; i = 0, 1, . . . , k$, and (in the
case $k = 0$) $L(x)_p \leq L(\varphi)_p$, where
$$L(x)_p := \sup \left\{\left(\int^b_a|x(t)|^p dt \right)^{1/p}\; :\; a, b \in \textbf{R},\; |x(t)| > 0,\; t \in (a, b) \right\}$$
In particular, we solve this extremal problem for Sobolev classes and for bounded sets of the spaces of
trigonometric polynomials and splines. |
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