Mathematical Sciences: Complex Analysis and Harmonic Analysis

工作将针对在理论中出现的问题， 多复变函数和调和分析。 研究 这是先前调查的继续， 三个主题：卷积方程，插值问题和 由指数多项式生成的理想。 卷积和插值问题的背景 可以追溯到一个卷积的基本问题， 测度集为零的单函数 同时，系统是否可以通过 测度的傅里叶变换的联合谱。 工作 这些问题导致的情况，虽然正确， 很容易验证。 首席调查员将追查几个 具体的标准，已经有部分 成功 一个特殊的障碍导致了数论 一个问题，估计根之间的距离， 指数多项式 在这方面取得了一些进展， 可能导致延迟型偏微分的应用 方程 一个不同类型的卷积问题出现在 另一个背景是庞贝问题。 在其 基本形式，问题涉及未知数的积分 函数将某些几何形状作为形状（或多个形状） 是由刚性运动驱动的 无论能否恢复 从这些计算函数是一个深刻的问题， 只有在涉及到相当多的案件中， 对称性 首席研究员将他的工作扩展到 流形（对称空间）的设置。 他已经证明了 庞培问题的失败， 关于椭圆型方程的Neumann边值问题 偏微分方程 虽然有许多应用程序 的结果，一个条件是强加在流形-他们 必须是rank 1。 将努力放宽这一要求。 这项研究与层析成像有着密切的联系。 调查事务所

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.