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This article is cited in 2 scientific papers (total in 2 papers)
About Kolmogorov type of inequalities for periodic functions of two variables in $L_2$
M. Sh. Shabozova, M. O. Akobirshoevb a Tajik national University (Dushanbe)
b Technological University of
Tajikistan (Dushanbe)
Abstract:
Let $L_{2}:=L_{2}(Q), \, Q:=\{0 \leq x, y \leq 2\pi\}$ be the
Hilbert space of summable with square of functions$f(x,y)$ in $Q$
domain with norm
\begin{equation*}
\|f\|_{2}:=\|f\|_{L_{2}(Q)}:=\left\{\frac{1}{4\pi^{2}}\iint_{(Q)}|f(x,y)|^2dxdy\right\}^{1/2}
< \infty,
\end{equation*}
and $L_{2}^{(r,s)}(Q)$ is class of functions $f\in L_{2}$ whose
derivatives $f^{(k,l)}\in C(Q)$, а $f^{(r,l)}, \, f^{(k,s)}$ $(0\leq
k\leq r-1$, $0\leq l\leq s-1, \, r,s\geq 2, r,s\in\mathbb{N})$,
$f^{(r,s)}$ are sectionally continuous and $f^{(r,s)}\in L_{2}$. In
this paper was proved that for arbitrary function $f\in
L_{2}^{(r,s)}$ is hold the following sharp Kolmogorov type
inequality
\begin{equation*}
\|f^{(r-k,s-l)}\|_{L_2(Q)}\leq\|f\|^{kl/rs}_{L_2(Q)}\cdot\|f^{(r,0)}\|^{(1-\frac{k}{r})\frac{l}{s}}_{L_2(Q)}\cdot
\|f^{(0,s)}\|^{\frac{k}{r}(1-\frac{l}{s})}_{L_2(Q)}\cdot\|f^{(r,s)}\|^{(1-\frac{k}{r})(1-\frac{l}{s})}_{L_2(Q)}.
\end{equation*}
Also, the Kolmogorov type inequality was found for the best
approximation $\mathscr{E}_{m-1,n-1}(f^{(r-k,s-l)})_{2}$ of
intermediate derivatives $f^{(r-k,s-l)}$ of functions $f\in
L_{2}^{(r,s)}$ by trigonometric “angles” with form
\begin{equation*}
\mathscr{E}_{m-1,n-1}(f^{(r-k,s-l)})_{2}\leq
\end{equation*}
\begin{equation*}\displaystyle\leq\left(\mathscr{E}_{m-1,n-1}\left(f\right)_{2}\right)^{kl/rs}\cdot\left(\mathscr{E}_{m-1,n-1}\left(f^{(r,0)}\right)_{L_{2}}\right)^{\left(1-\frac{k}{r}\right)\frac{l}{s}}\cdot\end{equation*}
\begin{equation*}
\cdot\left(\mathscr{E}_{m-1,n-1}\left(f^{(0,s)}\right)_{2}\right)^{\frac{k}{r}\left(1-\frac{l}{s}\right)}\cdot\left(\mathscr{E}_{m-1,n-1}\left(f^{(r,s)}\right)_{2}\right)^{\left(1-\frac{k}{r}\right)\left(1-\frac{l}{s}\right)},
\end{equation*}
This obtained inequality was applied for the problems of joint
approximation and their application in $L_{2}$. The sharp values of
linear and Kolmogorov widths for some classes of functions were
calculated.
Keywords:
Kolmogorov's type of inequalities, generalized polynomial, quasipolynomial, the best approximation, quasiwidth.
Received: 18.04.2019 Accepted: 12.07.2019
Citation:
M. Sh. Shabozov, M. O. Akobirshoev, “About Kolmogorov type of inequalities for periodic functions of two variables in $L_2$”, Chebyshevskii Sb., 20:2 (2019), 348–365
Linking options:
https://www.mathnet.ru/eng/cheb775 https://www.mathnet.ru/eng/cheb/v20/i2/p348
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Abstract page: | 150 | Full-text PDF : | 51 | References: | 23 |
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