Mathematical Sciences: Complex Analysis and Harmonic Analysis

工作将针对在多复变量理论和调和分析中出现的问题。这项研究代表了以下三个主题的早期研究的继续：卷积方程、插值问题和由指数多项式生成的理想。卷积和内插问题的背景可以追溯到单个函数通过一组同时消失的测量进行卷积的基本问题，以及是否可以通过测量的傅里叶变换的联合谱来分析系统。对这些问题的研究导致了一些条件，尽管这些条件是正确的，但并不容易得到证实。首席调查员将追求几个已经取得部分成功的具体类型标准。一个特殊的障碍导致了一个数论问题，即估计指数多项式的根之间的距离。在这方面已经取得了一些进展，可能会使其应用于时滞型偏微分方程解。在其他情况下出现的另一种卷积问题是庞贝问题。在它的基本形式中，这个问题涉及到当一个或多个形状通过刚性运动移动时，未知函数在某个几何形状上的积分。能否从这些计算中恢复函数是一个深刻的问题，其解决方案只有在涉及相当对称的情况下才能知道。主要研究人员已将他的工作扩展到流形(对称空间)的设置。他证明了Pompeiu问题的失败问题与椭圆型偏微分方程的Neumann边值问题直接相关。虽然这个结果有许多应用，但对流形施加了一个条件--它们必须是1阶的。我们将努力放宽这一要求。这项研究与断层扫描研究有着密切的联系。

Work will be directed at problems arising in the theories of several complex variables and harmonic analysis. The research represents a continuation of earlier investigations following three themes: convolution equations, interpolation problems and ideals generated by exponential polynomials. The background for problems of convolution and interpolation can be traced to fundamental questions of convolutions of a single function by a collection of measures vanishing simultaneously, and whether the system can be analyzed through the joint spectrum of the measures' Fourier transforms. Work on such questions led to conditions which, while correct, are not easily verified. The principal investigator will pursue several concrete-type criteria on which there has already been partial success. One particular impediment leads to a number-theoretic question which estimates the distance between roots of exponential polynomials. Some progress has been made here which may lead to applications to delay-type partial differential equations. A different type of convolution problem which appears in other contexts is that of the Pompeiu problem. In its rudimentary form, the problem concerns integrals of an unknown function taken over some geometric shape as the shape (or shapes) are moved by rigid motions. Whether or not one can recover the function from these calculations is a deep question whose resolution is known only in cases involving considerable symmetry. The principal investigator has expanded his work to the setting of manifolds (symmetric spaces). He has shown that the question of failure of the Pompeiu problem is directly related to a Neumann boundary value problem for an elliptic partial differential equation. While there are many applications of the result, a condition is imposed on the manifolds - they must be of rank 1. Work will be done in easing this requirement. This research has close connections to tomographic investigations.