About Kolmogorov type of inequalities for periodic functions of two variables in $L_2$

M. Sh. Shabozov ; M. O. Akobirshoev

Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 348-365

Résumé

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.

Détail

Keywords: Kolmogorov's type of inequalities, generalized polynomial, quasipolynomial, the best approximation, quasiwidth.