Interpolating Blaschke Products: Stolz and Tangential Approach Regions

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• 作者：Daniel Girela ; Jose Angel Pelaez and Dragan Vukotic
• 刊名：Constructive Approximation
• 出版年：2008
• 出版时间：March, 2008
• 年：2008
• 卷：27
• 期：2
• 页码：203-216
• 全文大小：162 KB

文摘

A result of D.J. Newman asserts that a uniformly separated sequence contained in a Stolz angle is a finite union of exponential sequences. We extend this by obtaining several equivalent characterizations of such sequences. If the zeros of a Blaschke product B lie in a Stolz angle, then B¡é ? ApB^\prime\in A^p for all p < \frac 32p and it has recently been shown that this result cannot be improved. Also, Newman's result can be used to prove that if B is an interpolating Blaschke product whose zeros lie in a Stolz angle, then B¡é ? ?_{0 < p < 1}Hp ¨¬ ?_{0 < p < 2}ApB^\prime\in\bigcap_{0 . In this paper we prove that if the zeros of an interpolating Blaschke product lie in a disk internally tangent to the unit circle, then B¡é ? ?_{0 < p < 3/2}ApB^\prime\in \bigcap_{0 and we show that this cannot be improved. We also obtain sharp results concerning the membership in Bergman spaces of the derivative of an interpolating Blaschke product whose sequence of zeros lies in other regions internally tangent to the unit circle.