Geodesic
Approximation of functions of a complex variable by Fourier sums in orthogonal systems in $L_2$
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2020), pp. 65-72

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The sharp inequalities of Jackson-Stechkin type inequalities between the best approximation $E_{n-s-1}(f^{(s)}) (s=\overline{0,r}, r\in\mathbb{N})$ of successive derivatives $f^{(s)} (s=\overline{0,r}, r\in\mathbb{N})$ of analytic functions $f\in L_{2}(U)$ in the disk $U:=\left\{z: |z|1\right\}$ as for special module of continuity $\Omega_{m}$ of $m$th order satisfying the condition $$\Omega_{m}\left(f^{(r)},t\right)_{2}\leq\Phi(t), 01,$$ where $\Phi$ is give majorant and also for Peetre $\mathscr{K}$-functional satisfying the constraint $$\mathscr{K}_{m}\left(f^{(r)},t^{m}\right)\leq\Phi(t^{m}), 01,$$ were obtained.
Keywords: the generalized module of continuity, generalized translation operator, orthonormal system of functions, Jackson–Stechkin inequality, $\mathscr{K}$-functional. 
@article{IVM_2020_6_a7,
     author = {M. Sh. Shabozov and M. S. Saidusaynov},
     title = {Approximation of functions of a complex variable by {Fourier} sums in orthogonal systems in $L_2$},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {65--72},
     publisher = {mathdoc},
     number = {6},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2020_6_a7/}
}
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M. Sh. Shabozov; M. S. Saidusaynov. Approximation of functions of a complex variable by Fourier sums in orthogonal systems in $L_2$. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2020), pp. 65-72. http://geodesic.mathdoc.fr/item/IVM_2020_6_a7/