Mathematical Sciences: Complex Analysis: Computation and Control.

这个项目将专注于数学分析的四个领域。我们将在位势理论中使用的二维调和测度的估计以及在测量解析函数的增长性方面做工作。目前已知某一扇区内满足所有半径的圆盘中曲线的调和度量的估计值。最大扇区的宽度仍有待确定。在这一过程中，将结合理论和数字努力。关于使用有界全纯函数的最优化问题的工作将继续下去。这些调查已经进行了几年。它们源于最优控制中的某些工程问题，涉及到通过定义在圆上的有界函数或连续函数来寻找对给定函数的最佳逼近，这些函数具有对内部的解析扩展。该项目的第三个目标，也是最近的目标，将涉及计算保角映射的方法。对于具有光滑边界的区域的这种计算和映射，存在许多技术，这些技术通常具有快速收敛的数值逼近。本文研究了具有角点的区域映射的逼近问题。最近的进展是令人鼓舞的，尽管重复的近似过程在几次通过后显示出退化。将努力挑选出最好的边界曲线，目前的方法是可行的。这些曲线很可能属于限制弯曲的弦弧曲线。最后，从更具理论性的角度，研究了圆盘的最大理想空间的Gleason部分上不变的内函数的存在性和性质。虽然一般理论表明存在这样的内部函数，但关于这类函数中是否存在Blaschke积却鲜为人知。与这项工作相关的是这类产品的零分布的刻画。除了应用到位势理论之外，这项工作还增加了关于偏微分方程数值逼近格式的基本知识。

This project will focus on four areas of mathematical analysis. Work will be done on estimating two-dimensional harmonic measure which is used in potential theory and in measuring growth of analytic functions. At the present time estimates are known for the harmonic measure of curves in a disc meeting all radii in some sector. The width of the maximal sector remains to be determined. Theoretical and numerical efforts will be combined in this process. Work will continue on problems of optimization using bounded holomorphic functions. These investigations have been underway for several years. They arise from certain engineering problems in optimal control and involve finding best approximations to given functions by bounded or continuous functions defined on a circle which have analytic extensions to the interior. A third, and more recent, goal of this project will involve methods for the computation of conformal maps. There are many techniques extant for such computations and mappings of domains with smooth boundaries that generally have numerical approximants which converge rapidly. The present work is concerned with approximation involving maps of domains with corners. Recent progress has been encouraging although repetitions of the approximation process shows degradation after several passes. Efforts will be made to single out the best boundary curves for which current methods are practical. It is likely that these curves will fall into the class known as chord-arc curves which have restricted bending. Finally, in a more theoretical vein, studies will be made into the existence and nature of inner functions which are constant on Gleason parts of the maximal ideal space of a disc. While general theory indicates existence of such inner functions there is little known as to whether there are Blaschke products in this class. Related to this work is a characterization of the zero-distribution of such products. In addition to applications to potential theory, this work adds to fundamental knowledge regarding numerical approximation schemes for partial differential equations.