Applications of Fourier analysis to convex geometry and geometric tomography

傅里叶分析在凸几何和几何断层扫描中的应用

• 批准号: RGPIN-2014-03874
• 负责人: Yaskin, Vladyslav
• 金额: $ 1.31万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=669259
• 关键词: Applications; Fourier; analysis; convex; geometry

The main theme of the proposed research project is centered around the study of sections, projections, volumes of convex and star bodies using methods of Fourier analysis. The area dealing with the retrieval of information about convex or star bodies based on data obtained from various measurements is known as geometric tomography. The book "Geometric Tomography" by R. J. Gardner gives an excellent overview of known results, techniques, and open problems in this area. The area lies at the intersection of several branches of mathematics: functional analysis, harmonic analysis, convex geometry, differential geometry, integral geometry etc. This explains the diversity of methods employed in geometric tomography. Our approach is based on techniques of harmonic analysis. Fundamental discoveries in this direction were made by A. Koldobsky at the end of the last century, who applied the Fourier transform to the study of sections of convex bodies (see the book "Fourier Analysis in Convex Geometry"). Since then such methods have been successfully used to solve a variety of problems in geometric tomography, convex geometry, geometric functional analysis etc. The goal of the proposed research project is to find new applications of such techniques. Even though this project concerns purely theoretical aspects of geometric tomography, results and methods from this area find applications in many scientific fields. One of the most well-known applications is computer assisted tomography, which allows to generate images from X-rays of human patients.

提出的研究项目的主题是围绕使用傅里叶分析方法研究截面，投影，凸体和星体的体积。基于从各种测量中获得的数据，处理关于凸体或星体的信息检索的领域称为几何层析成像。R. J. Gardner的《几何断层扫描》一书对该领域的已知结果、技术和开放问题进行了极好的概述。该领域位于几个数学分支的交叉点：泛函分析、调和分析、凸几何、微分几何、积分几何等。这解释了几何层析成像方法的多样性。我们的方法是基于谐波分析技术。在这个方向上的基本发现是由A. Koldobsky在上世纪末做出的，他将傅里叶变换应用于凸体截面的研究（参见《凸几何中的傅里叶分析》一书）。从那时起，这种方法已成功地用于解决几何层析成像、凸几何、几何泛函分析等各种问题。拟议的研究项目的目标是找到这些技术的新应用。尽管该项目涉及几何层析成像的纯理论方面，但该领域的结果和方法在许多科学领域都有应用。最著名的应用之一是计算机辅助断层扫描，它可以从人类患者的x射线中生成图像。