Positivity, Inverse Problems, and Operator Theory

该项目的重点是从函数和复杂数学分析的角度研究反问题的数学方面。统计学、光谱分析、X射线晶体学、断层摄影术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，矩问题形成了一个有着大约两个世纪迷人历史的成熟的研究领域。该项目的重点是从远程测量（例如电场或重力场）的形状和体积重建中产生的特定力矩问题。该项目的一个重要组成部分是研究与重建算法有关的最佳近似和误差范围。首席研究员以前在类似的二维问题上的成功研究保证了他试图将视野扩展到三维或更多维的积极结果。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对于当前泛函分析、函数论和数值数学的研究，以及应用数学和工程的某些具体分支都有重要意义。

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.