Mean-square approximation of functions of a complex variable by Fourier sums in orthogonal systems

M. Sh. Shabozov ; M. S. Saidusajnov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 2, pp. 258-272

Résumé

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}), \ \ 01, \end{equation*} where $\Psi$ is some increasing majorant and $\Psi(0)=0$.

Détail

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