Topics in Analysis

分析主题

• 批准号: 0601025
• 负责人: Charles Fefferman
• 金额: $ 73.89万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0601025&HistoricalAwards=false
• 关键词: Topics; Analysis

Fefferman showed how to decide whether a given real-valued function on a compact subset of R^n extends to a C^m function on all of R^n. This answers a classical problem of Whitney. Together with Bo'az Klartag, Fefferman gave an effective version of this work, thus providing an asymptotically efficient algorithm to compute a C^m function with controlled C^m norm, approximating a given function on a large finite set. Stein's research involves the relation and interplay between algebras of pseudo-differential and singular integral operators arising in the context of several complex variables and for sub-elliptic differential operators. He also deals with the problem of extending to the discrete setting the theory of singular Radon transforms. Together with B. Klartag, Fefferman found an algorithm to compute a smooth surface passing through (or near to) many given points in space, whenever such a surface exists. Fefferman's work is related to practical problems of computational geometry, arising in computer-aided manufacturing.Stein's research advances Fourier analysis, which is a fundamental tool in many branches of science and technology. It is used e.g. to study waves of all kinds, digital and analog signals and medical imaging.

费曼展示了如何判断一个给定的实值函数在R^n的紧致子集上是否可以扩展为一个C^m函数在所有R^n上。这回答了惠特尼的一个经典问题。与Bo'az Klartag一起，Febriman给出了这项工作的一个有效版本，从而提供了一个渐近有效的算法来计算具有受控C^m范数的C^m函数，近似于一个大的有限集上的给定函数。斯坦的研究涉及代数之间的关系和相互作用的伪微分和奇异积分算子的背景下产生的几个复杂的变量和次椭圆微分算子。 他还处理的问题延伸到离散设置的理论奇异拉东变换。和B一起。Klartag，February man发现了一个算法来计算一个光滑的表面通过（或接近）许多给定的点在空间中，只要这样的表面存在。 费曼的工作与计算机辅助制造中出现的计算几何的实际问题有关。斯坦的研究促进了傅立叶分析，这是许多科学和技术分支的基本工具。 它用于研究各种波、数字和模拟信号以及医学成像。