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Zapiski Nauchnykh Seminarov LOMI, 1985, Volume 141, Pages 72–99
(Mi znsl4167)
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This article is cited in 1 scientific paper (total in 1 paper)
A constructive description of Hölder classes on closed Jordan curves
N. A. Shirokov
Abstract:
Let $\Gamma$ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, let $a\in\operatorname{int}\Gamma$, and $\mathcal {R}_n(a)$ be the set of rational functions of degree $le n$, having a pole perhaps only at the point $a.$ Let $\Lambda^{\alpha}(\Gamma)$, $0<\alpha<1,$ be the Hölder class on $\Gamma.$ One constructs a system of weights $\gamma_n(z)>0$ on $\Gamma$ such that $f\in\Lambda^{\alpha}(\Gamma)$ if and only if for any nonnegative integer $n$ there exists a function $R_n$, $R_n\in\mathcal {R}_n(a)$ such that $|f(z)-R_n(z)|\le c_f\cdot\gamma_n(z)$, $z\in\Gamma.$
It is proved that the weights $\gamma_n$ cannot be expressed simply in terms in terms of $\rho^+_{1/n}(z)$ and $\rho^-_{1/n}(z)$, the distances to the level lines of the moduli of the conformal mappings of $\operatorname{ext}\Gamma$ and $\operatorname{int}\Gamma$ on $\mathbb C\backslash\mathbb D.$
Citation:
N. A. Shirokov, “A constructive description of Hölder classes on closed Jordan curves”, Investigations on linear operators and function theory. Part XIV, Zap. Nauchn. Sem. LOMI, 141, "Nauka", Leningrad. Otdel., Leningrad, 1985, 72–99; J. Math. Sci., 37:5 (1937), 1306–1322
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https://www.mathnet.ru/eng/znsl4167 https://www.mathnet.ru/eng/znsl/v141/p72
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