Estimates of the Values of $n$-Widths of Classes of Analytic Functions in the Weight Spaces $H_{2,\gamma}(D)$

S. B. Vakarchuk

Matematičeskie zametki, Tome 108 (2020) no. 6, pp. 803-822

Résumé

In a simply connected bounded domain $D\subset\mathbb C$ with rectifiable Jordan boundary $\partial D$, we study the classes $H_{2,\gamma}(D;\Omega_k,\Phi)$, $k\in\mathbb N$, consisting of analytic functions $f\in H_{2,\gamma}(D)$ in $D$ each of which, for any $t\in(0,1)$, satisfies the condition $\Omega_k(f,t)\le\Phi(t)$. Here $\Omega_k(f)$ is the generalized modulus of continuity of $k$th order in $H_{2,\gamma}(D)$ and $\Phi$ is a majorant. For these classes, we find upper and lower bounds for various $n$-widths, as well as upper bounds for the moduli of Fourier coefficients. We obtain a constraint on the majorant $\Phi$ under which the exact values of these extremal characteristics can be calculated. In the case of the unit disk, similar results are obtained for classes of analytic functions whose definitions include the Hadamard compositions $\mathscr D(\mathscr B_m,f)$ in addition to $\Omega_k(f)$ and $\Phi$. Concrete realizations of some obtained exact results are presented.

Détail

Keywords: weight function, orthogonal system of polynomials, generalized modulus of continuity, Fourier series, $n$-width.
Mots-clés : majorant, Fourier coefficient, Hadamard composition