On the Bernstein - Walsh-type lemmas in regions of the complex plane
Let $G \subset C$ be a finite region bounded by a Jordan curve $L := \partial G,\quad \Omega := \text{ext} \; \overline{G}$ (respect to $\overline{C}$), $\Delta := \{z : |z| > 1\}; \quad w = \Phi(z)$ be the univalent conformal mapping of $\Omega$ ont $\Phi$ normalized by $\Phi(\infty) = \inf...
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| Дата: | 2011 |
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| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2011
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/2717 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| Резюме: | Let $G \subset C$ be a finite region bounded by a Jordan curve $L := \partial G,\quad \Omega := \text{ext} \; \overline{G}$ (respect to $\overline{C}$), $\Delta := \{z : |z| > 1\}; \quad w = \Phi(z)$ be the univalent conformal mapping of $\Omega$ ont $\Phi$ normalized by $\Phi(\infty) = \infty,\quad \Phi'(\infty) > 0$.
Let $A_p(G),\; p > 0$, denote the class of functions $f$ which are analytic in $G$ and satisfy the condition
$$||f||^p_{A_p(G)} := \int\int_G |f(z)|^p d \sigma_z < \infty,\quad (∗)$$
where $\sigma$ is a two-dimensional Lebesque measure.
Let $P_n(z)$ be arbitrary algebraic polynomial of degree at most $n$. The well-known Bernstein – Walsh
lemma says that
$$P_n(z)k ≤ |\Phi(z)|^{n+1} ||P_n||_{C(\overline{G})}, \; z \in \Omega. \quad (∗∗)$$
Firstly, we study the estimation problem (∗∗) for the norm (∗). Secondly, we continue studying the
estimation (∗∗) when we replace the norm $||P_n||_{C(\overline{G})}$ by $||P_n||_{A_2(G)}$
for some regions of complex plane. |
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