TY - JOUR

T1 - A constructive description of Hölder classes on closed Jordan curves

AU - Shirokov, N. A.

PY - 1987/6

Y1 - 1987/6

N2 - Let Γ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, let a ε int Γ, and let Rn(a) be the set of rational functions of degree ≤n, having a pole perhaps only at the point a. Let Λα(Γ), 0 < α < 1, be the Hölder class on Γ. One constructs a system of weights γn(z) > 0 on Γ such that f∈Λα(Γ) if and only if for any nonnegative integer n there exists a function Rn, Rn ε Rn(a) such that |f(z) - Rn(z)| ≤ cf·γn(z), z ε Γ. It is proved that the weights γn cannot be expressed simply in terms of ρ1+/n(z) and ρ1-/n(z), the distances to the level lines of the moduli of the conformal mappings of ext Γ and int Γ on {Mathematical expression}.

AB - Let Γ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, let a ε int Γ, and let Rn(a) be the set of rational functions of degree ≤n, having a pole perhaps only at the point a. Let Λα(Γ), 0 < α < 1, be the Hölder class on Γ. One constructs a system of weights γn(z) > 0 on Γ such that f∈Λα(Γ) if and only if for any nonnegative integer n there exists a function Rn, Rn ε Rn(a) such that |f(z) - Rn(z)| ≤ cf·γn(z), z ε Γ. It is proved that the weights γn cannot be expressed simply in terms of ρ1+/n(z) and ρ1-/n(z), the distances to the level lines of the moduli of the conformal mappings of ext Γ and int Γ on {Mathematical expression}.

UR - http://www.scopus.com/inward/record.url?scp=34250109551&partnerID=8YFLogxK

U2 - 10.1007/BF01327040

DO - 10.1007/BF01327040

M3 - Article

AN - SCOPUS:34250109551

VL - 37

SP - 1306

EP - 1322

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 5

ER -