A~constructive description of H\"older classes on closed Jordan curves

N. A. Shirokov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XIV, Tome 141 (1985), pp. 72-99

Résumé

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 $\Lambda^{\alpha}(\Gamma)$, $0\alpha1,$ 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 $\mathbb C\backslash\mathbb D.$