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Chebyshevskii Sbornik, 2019, Volume 20, Issue 2, Pages 348–365
DOI: https://doi.org/10.22405/2226-8383-2018-20-2-348-365
(Mi cheb775)
 

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)
Full-text PDF (692 kB) Citations (2)
References:
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
Document Type: Article
UDC: 517.5
Language: Russian
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
Citation in format AMSBIB
\Bibitem{ShaAko19}
\by M.~Sh.~Shabozov, M.~O.~Akobirshoev
\paper About Kolmogorov type of inequalities for periodic functions of two variables in $L_2$
\jour Chebyshevskii Sb.
\yr 2019
\vol 20
\issue 2
\pages 348--365
\mathnet{http://mi.mathnet.ru/cheb775}
\crossref{https://doi.org/10.22405/2226-8383-2018-20-2-348-365}
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  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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