Mean-squared approximation of some classes of complex variable functions by Fourier series in the weighted Bergman space $B_{2,\gamma}$

M. Sh. Shabozov ; M. S. Saidusainov

Čebyševskij sbornik, Tome 23 (2022) no. 1, pp. 167-182

Résumé

The article considers extremal problems of mean-square approximation of functions of a complex variable, regular in the domain $\mathscr{D}\subset\mathbb{C}$, by Fourier series orthogonal in the system of functions $\{\varphi_{k}(z)\}_{k=0}^{\infty}$ in $\mathscr{D}$ belonging to the weighted Bergman space $B_{2,\gamma}$ with finite norm \begin{equation*} \|f\|_{2,\gamma}:=\|f\|_{B_{2,\gamma}}=\left(\frac{1}{2\pi}\iint\limits_{(\mathscr{D})}\gamma(|z|)|f(z)|^{2}d\sigma\right)^{1/2},\end{equation*} where $\gamma:=\gamma(|z|)\geq 0$ is a real integrable function in the domain $\mathscr{D}$, and the integral is understood in the Lebesgue sense, $d\sigma:=dxdy$ is an element of area. The formulated problem is investigated in more detail in the case when $\mathscr{D}$ is the unit disc in the space $B_{2,\gamma_{\alpha,\beta}}, \gamma_{\alpha,\beta}=|z|^{\alpha}(1-|z|)^{\beta}, \alpha,\beta>-1$ – Jacobi weight. Sharp Jackson-Stechkin-type inequalities that relate the value of the best mean-squared polynomial approximation of $f\in \mathcal{B}_{2,\gamma_{\alpha,\beta}}^{(r)}$ and the Peetre $\mathscr{K}$-functional were proved. In case when $\gamma_{\alpha,\beta}\equiv 1$ we will obtain the earlier known results.

Détail

Keywords: Fourier's sum, mean-squared approximation, upper bound best approximation, Peetre $\mathscr{K}$-functional.