M.Sh. Shabozov, M.S. Saidusainov. Mean-square approximation of functions of a complex variable by Fourier sums in orthogonal systems ... P. 258-272

Assume that $\mathcal{A}(U)$ is the set of functions analytic in the disk $U:=\{z: |z|<1\}$, $L_2^{(r)}:=L_2^{(r)}(U)$ for $r\in\mathbb{N}$ is the class of functions $f\in\mathcal{A}(U)$ such that $f^{(r)}\in L_2^{(r)}$, and $W^{(r)}L_2$ is the class of functions $f\in L_2^{(r)}$ satisfying the constraint $\|f^{(r)}\|\leq 1$. We find exact values for mean-square approximations of functions $f\in W^{(r)}L_2$ and their successive derivatives $f^{(s)}$ ($1\leq s\leq r-1$, $r\geq 2$) in the metric of the space $L_2$. A similar problem is solved for the class $W_2^{(r)}(\mathscr{K}_{m},\Psi)$ ($r\in\mathbb{Z}_{+}$, $m\in\mathbb{N}$) of functions $f\in L_2^{(r)}$ such that the $\mathscr{K}$-functional of their $r$th derivative satisfies the condition \begin{equation*} \mathscr{K}_{m}\left(f^{(r)},t^{m}\right)\leq\Psi(t^{m}), \ \ 0<t<1, \end{equation*}
where $\Psi$ is some increasing majorant and $\Psi(0)=0$.

Keywords: generalized modulus of continuity, generalized translation operator, orthonormal system, Jackson-Stechkin inequality, $\mathscr{K}$-functional

Received February 28, 2019

Revised May 24, 2019

Accepted May 27, 2019

Mirgand Shabozovich Shabozov, Dr. Sci. (Phys.-Math.), Prof., Tajik National University, Dushanbe, 734025 Republic of Tajikistan; University of Central Asia, Dushanbe, SPCE, 734013 Republic of Tajikistan, e-mail: shabozov@mail.ru

Mukim Saidusaynovich Saidusaynov, Cand. Sci. (Phys.-Math.), Tajik National University, Dushanbe, 734025 Republic of Tajikista; University of Central Asia, Dushanbe, SPCE, 734013 Republic of Tajikistan, e-mail: smuqim@gmail.com

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Cite this article as: M.Sh.Shabozov, M.S.Saidusainov. Mean-square approximation of functions of a complex variable by Fourier sums in orthogonal systems, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 2, pp. 258–272.