Topics in Analysis

Fefferman展示了如何确定一个给定的实值函数在R^n的紧子集上是否可以扩展到整个R^n上的C^m函数。这回答了惠特尼的一个经典问题。与Bo'az Klartag一起，Fefferman给出了这项工作的一个有效版本，从而提供了一个渐进有效的算法来计算具有控制C^m范数的C^m函数，近似于一个大有限集合上的给定函数。Stein的研究涉及在复数变量和次椭圆微分算子的情况下产生的伪微分算子和奇异积分算子代数之间的关系和相互作用。他还讨论了将奇异Radon变换理论推广到离散环境的问题。费费曼和B. Klartag一起发现了一种算法，可以计算一个光滑的表面，无论这个表面是否存在，它都会穿过（或接近）空间中许多给定的点。费费曼的工作与计算机辅助制造中产生的计算几何的实际问题有关。Stein的研究促进了傅里叶分析，这是许多科学和技术分支的基本工具。例如，它被用于研究各种波、数字和模拟信号以及医学成像。

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.