Positivity, Inverse Problems, and Operator Theory

该项目以反问题的数学方面为中心，从函数和复杂数学分析的角度进行研究。统计学、光谱分析、x射线晶体学、断层扫描为这些问题提供了模型和经典的例子。更具体地说，大多数现代反问题都与质量分布的（幂、傅立叶、小波）矩有关。反过来，矩问题形成了一个成熟的研究领域，有着大约两个世纪的迷人历史。该项目侧重于从远距离测量（例如电场或引力场）中产生的形状和体积重建中的特定力矩问题。该项目的一个重要组成部分是研究与重建算法相关的最佳近似值和误差范围。首席研究员之前在类似二维问题上的成功研究保证了他在尝试将视野扩展到三维或更多维度时取得积极成果。他将与几位纯数学和应用数学专家以及研究生和年轻的研究助理合作。他们的共同努力和成果将对当前泛函分析、泛函理论和数值数学以及应用数学和工程的一些特定分支的研究具有重要意义。

The project is centered on the study of mathematical aspects of inverse problems, from the point of view of functional and complex mathematical analysis. Statistics, spectral analysis, X-ray crystallography, tomography offer models and classical examples of such questions. More specifically, most of the modern inverse problems are related to (power, Fourier, wavelet) moments of mass distributions. In their turn, moment problems form a wellestablished field of research with a fascinating history of abouttwo centuries. The project focuses on specific moment problems arising in shape and volume reconstruction from distant measurements (for instance of electric or gravitational fields). An important component of the project is the study of the best approximation and the error bounds related to reconstruction algorithms.The success of previous researches of the principal investigator on similar two dimensional questions guarantee positive results in his attempt to expand his horizon to three or more dimensions. He will collaborate with several experts in pure and applied mathematics, as well as with graduate students andyoung research assistants. Their combined efforts and results will be significant for present studies in functional analysis, function theory and numerical mathematics, as well as for some specific branches of appliedmathematics and engineering.