Mathematical Sciences: Interpolation by Meromorphic Matrix Functions

工作要做的这个项目结合了数学理论的封闭 多值矩阵值函数 这些研究的潜在动机在于预期在系统理论和泛函分析中的重要应用。这些研究可以从插值理论和亚纯矩阵函数的角度来看待，也可以作为闭Riemann曲面上复向量丛的Weil因子方法和自同构因子方法之间的对应关系的明确工作。 前者的目的是构造在闭黎曼曲面上插值给定点的亚纯矩阵值函数。 在后者中，一个地方的解决方案，以形成一个全纯向量丛，并要求的条件，保证丛是平凡的。这就是Weil的工作进入的地方。 在这个项目中所描述的工作出现在系统分析中，其中一个感兴趣的是从所有成员满足相同条件的类中选择最优函数。 条件可以作为函数的规定逐点值给出。 因此，在确定最优函数之前，必须首先解决插值问题，对所有这些函数进行分类。

Work to be done on this project combines the mathematical theories of closed multiple-valued, matrix valued functions. The underlying motivation for the studies lies in the expected significant applications to systems theory and functional analysis. The research can be viewed from the point of view of interpolation theory and meromorphic matrix functions, or as explicit work on the correspondence between the Weil divisor approach and the factor of automorphy approach to complex vector bundles over closed Riemann surfaces. In the former one wants to construct meromorphic matrix valued functions interpolating given points on a closed Riemann surface. In the latter, one takes local solutions to form a holomorphic vector bundle and asks for conditions which guarantee that the bundle be trivial. This is where Weil's work enters. The work described in this project arises in systems analysis where one is interested in selecting optimal functions from classes all of whose members satisfy the same conditions. The conditions may be given as prescribed pointwise values of the functions. It then becomes essential that one first solve the interpolation problem to classify all such functions before establishing the optimal ones.