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On the convergence of the Bieberbach polynomials in regions with corners. (English) Zbl 0645.30002

Zum von der Jordankurve \(L\) berandeten Gebiet \(G\ni 0\) sei \(f_ 0(z)\) die schlichte konforme Abbildung auf \(| w| <r_ 0\) mit \(f_ 0(0)=0\), \(f_ 0'(0)=1\); dazu \(\pi_ n(z)\) die Bieberbachschen Polynome. In der Arbeit wird das Verhalten der Abweichung \[ \max | f_ 0(z)- \pi_ n(z)| \quad bei\quad z\in \bar G \] für \(n\to \infty\) untersucht, wenn \(L\) stückweise glatt ist, also jedenfalls Ecken haben darf. Diese Abweichung wurde früher bei glattem \(L\) von Keldych, Suetin, Mergelyan u.a. untersucht.
Hauptergebnis: Ist \(\lambda\pi\) \((0<\lambda <2)\) der kleinste Außenwinkel bei den Ecken von \(L\), dann ist diese Abweichung \(\leq \text{const}\cdot n^{-\gamma}\) für alle \(\gamma\) mit \[ \gamma <\min (\lambda /(2-\lambda),1/2). \] Diese Schranke für \(\gamma\) ist zumindest für \(\lambda\leq 2/3\) nicht zu vergrößern. Der Beweis fußt u.a. auf Ansätzen von V. V. Andrievskii [Ukr. Math. J. 35, 233–236 (1983; Zbl 0536.30006)] und benutzt quasikonforme Spielgelungen.
Außerdem gibt es Abschätzungen für \(\gamma\), wenn für \(f_ 0(z)\) und die inverse Abbildung Lipschitzabschätzungen bekannt sind.
Sorgfältige numerische Experimente zur Ergänzung.
Reviewer: R.Kühnau

MSC:

30C30 Schwarz-Christoffel-type mappings
30E10 Approximation in the complex plane
30C70 Extremal problems for conformal and quasiconformal mappings, variational methods

Citations:

Zbl 0536.30006
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References:

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