Topics in Analysis

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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.