Mathematical Sciences: Complex Analysis and Harmonic Analysis

该项目继续数学研究，重点是 复分析法在卷积分析中的应用 方程、积分几何和代数系统 方程 空间中插值问题的早期工作 整个功能自然会引发有关 多维卷积方程 一个实际上是 对反卷积的问题感兴趣。 许多物理 例如换能器和传感器的可实现系统可以 用分布的卷积来建模。 这产生 用于估计诸如图像，辐射， 数据以卷积加噪声的形式给出 该项目的重建过程包括 找到系统的方法来构造去卷积器。 在 在进行这项工作时，出现了几个数学问题， 被视为不同的目标。 其中包括寻找 度和高度的代数Bezout方程，获得 剩余电流的新性质和表示 （多维留数）与积分几何问题 与Pompieu问题有关。 庞贝问题实际上 涵盖了大量的问题，关于积分的一个 在各种全等配置上的固定函数。 如果 在一类全等集上，整数等于零，一个是 我想知道函数是否为零。 这项工作的大部分是在几个框架内进行的， 复杂变量 解析函数理论与 谐波分析占主导地位。 多种应用， 系统理论与工程。

This project continues mathematical research focusing on complex analytic methods applied to the analysis of convolution equations, integral geometry and systems of algebraic equations. Earlier work on interpolation problems in spaces of entire functions has led naturally to questions concerning multidimensional convolutions equations. One is actually interested in the question of deconvolution. Many physical realizable systems such as transducers and sensors can be modeled as convolution with distributions. This gives rise to a scheme to estimate signals such as images, radiation, acoustics, etc. The data are given as a convolution plus noise and the reconstruction process in this project consists of finding systematic ways of constructing deconvolvers. In pursuing this work, several mathematical problems arise which are treated as distinct objectives. They include finding the degrees and height in the algebraic Bezout equation, obtaining new properties and representations of residue currents (multidimensional residues) and questions in integral geometry related to the Pompieu problem. The Pompieu problem actually covers a vast array of questions concerning integrals of a fixed function over various congruent configurations. If the integtrals equal zero on a class of congruent sets, one is interested in knowing whether or not the function must be zero. Most of this work is set within the framework of several complex variables. Techniques of analytic function theory and harmonic analysis are predominant. Diverse applications to systems theory and engineering are expected.