There have been many elegant results discussing the approximation proper- ties of the Bieberbach polynomials. However, very few papers investigated the approxi- mation properties of the extremal polynomials over Jorda...There have been many elegant results discussing the approximation proper- ties of the Bieberbach polynomials. However, very few papers investigated the approxi- mation properties of the extremal polynomials over Jordan curves. In the present paper, some results on a class of extremal polynomials over C1+αsmooth Jordan curves are obtained.展开更多
Let G C C be a simply connected domain whose boundary L ：= G is a Jordan curve and 0 ∈ G. Let w = φ（z） be the conformal mapping of G onto the disk B（0, r0） ：= {w ： ｜w｜ 〈 r0）, satisfying φ0（0） = 0, φ＇...Let G C C be a simply connected domain whose boundary L ：= G is a Jordan curve and 0 ∈ G. Let w = φ（z） be the conformal mapping of G onto the disk B（0, r0） ：= {w ： ｜w｜ 〈 r0）, satisfying φ0（0） = 0, φ＇t（0） = 1. We consider the following extremal problem for p 〉 0：∫∫G｜φ＇（z）-P＇n（z）｜Pdσz→min in the class of all polynomials Pn（z） of degree not exceeding n with Pn（0） = 0, P＇n （0）=- 1. The solution to this extremal problem is called the p-Bieberbach polynomial of degree n for the pair （G, 0）. We study the uniform convergence of the p-Bieberbach polynomials Bn,p（z） to the φ（z） on G^- with interior and exterior zero angles determined depending on the properties of boundary arcs and the degree of their ＂touch＂.展开更多
基金Supported by the National Science Foundation of China (19771006)
文摘There have been many elegant results discussing the approximation proper- ties of the Bieberbach polynomials. However, very few papers investigated the approxi- mation properties of the extremal polynomials over Jordan curves. In the present paper, some results on a class of extremal polynomials over C1+αsmooth Jordan curves are obtained.
文摘Let G C C be a simply connected domain whose boundary L ：= G is a Jordan curve and 0 ∈ G. Let w = φ（z） be the conformal mapping of G onto the disk B（0, r0） ：= {w ： ｜w｜ 〈 r0）, satisfying φ0（0） = 0, φ＇t（0） = 1. We consider the following extremal problem for p 〉 0：∫∫G｜φ＇（z）-P＇n（z）｜Pdσz→min in the class of all polynomials Pn（z） of degree not exceeding n with Pn（0） = 0, P＇n （0）=- 1. The solution to this extremal problem is called the p-Bieberbach polynomial of degree n for the pair （G, 0）. We study the uniform convergence of the p-Bieberbach polynomials Bn,p（z） to the φ（z） on G^- with interior and exterior zero angles determined depending on the properties of boundary arcs and the degree of their ＂touch＂.
