2019
Том 71
№ 6

All Issues

Özkartepe N. P.

Articles: 2
Article (English)

Interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces. I

Abdullayev F. G., Özkartepe N. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2016. - 68, № 10. - pp. 1365-1379

We study the order of the height of the modulus of arbitrary algebraic polynomials with respect to the weighted Lebesgue space, where the contour and the weight functions have some singularities.

Article (English)

On the improvement of the rate of convergence of the generalized Bieberbach polynomials in domains with zero angles

Abdullayev F. G., Özkartepe N. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2012. - 64, № 5. - pp. 582-596

Let $\mathbb{C}$ be the complex plane, let $\overline{\mathbb{C}} = \mathbb{C} \bigcup \{\infty\}$, let $G \subset \mathbb{C}$ be a finite Jordan domain with $0 \in G$, let $L := \partial G$, let $\Omega := \overline{\mathbb{C}} \ \overline{G}$, and let $w = \varphi(z)$ be the conformal mapping of $G$ onto a disk $B(0, \rho) := \{w : \; |w | < \rho_0\}$ normalized by $\varphi(0) = = 0,\; \varphi'(0) = 1$, where $\rho_0 = \rho_0 (0, G)$ is the conformal radius of $G$ with respect to 0. Let $\varphi \rho(z) := \int^z_0 [\varphi'(\zeta)]^{2/p}d\zeta$ and let $\pi_{n,p}(z)$ be the generalized Bieberbach polynomial of degree $n$ for the pair $(G, 0)$ that minimizes the integral $\int\int_G|\varphi'(z) - P'_n(z)|^p d \sigma_z$ in the class of all polynomials of degree $\text{deg} P_n \leq n$ such that $P_n(0) = 0$ and $P'_n(0) = 1$. We study the uniform convergence of the generalized Bieberbach polynomials $\pi_{n,p}(z)$ to $\varphi \rho(z)$ on $\overline{G}$ with interior and exterior zero angles determined depending on properties of boundary arcs and the degree of their tangency. In particular, for Bieberbach polynomials, we obtain better estimates for the rate of convergence in these domains.