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This article is cited in 1 scientific paper (total in 2 paper)
Szegő theorem, Carathéodory domains, and boundedness of calculating functionals
F. G. Abdullaeva, A. A. Dovgosheyb a University of Mersin
b Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences
Abstract:
Suppose that $G$ is a bounded simply connected domain on the plane with boundary $\Gamma$, $z_0\in G$, $\omega$ is the harmonic measure with respect to $z_0$, on $\Gamma$, $\mu$ is a finite Borel measure with support $\operatorname{supp}(\mu)\subseteq\Gamma$, $\mu_a+\mu_s$ is the decomposition of $\mu$ with respect to $\omega$, and $t$ is a positive real number. We solve the following problem: for what geometry of the domain $G$ is the condition
$$
\int\ln\biggl(\frac{d\mu_a}{d\omega}\biggr)\,d\omega=-\infty
$$
equivalent to the completeness of the polynomials in$L^t(\mu)$ or to the unboundedness of the calculating functional $p\to p(z_0)$, where $p$ is a polynomial in $L^t(\mu)$? We study the relationship between the densities of the algebras of rational functions in $L^t(\mu)$ and $C(\Gamma)$. For $t=2$, we obtain a sufficient criterion for the unboundedness of the calculating functional in the case of finite Borel measures with support of an arbitrary geometry.
Received: 26.09.2002
Citation:
F. G. Abdullaev, A. A. Dovgoshey, “Szegő theorem, Carathéodory domains, and boundedness of calculating functionals”, Mat. Zametki, 77:1 (2005), 3–15; Math. Notes, 77:1 (2005), 3–14
Linking options:
https://www.mathnet.ru/eng/mzm2464https://doi.org/10.4213/mzm2464 https://www.mathnet.ru/eng/mzm/v77/i1/p3
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Abstract page: | 391 | Full-text PDF : | 188 | References: | 41 | First page: | 2 |
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