Approximation of functions of a complex variable by Fourier sums in orthogonal systems in $L_2$

M. Sh. Shabozov ; M. S. Saidusaynov

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2020), pp. 65-72

Résumé

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.

Détail

Keywords: the generalized module of continuity, generalized translation operator, orthonormal system of functions, Jackson–Stechkin inequality, $\mathscr{K}$-functional.