Carleson's Theorem in Analysis, Scattering, and Ergodic Theory

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Wave packet analysis has applications to various fields within mathematical analysis: scattering theory, classical Fourier analysis, ergodic Theory. This project will broaden the use of wave packet analysis in these areas and thus lead to deeper understanding of these areas.Particular problems considered in this project are a natural nonlinear variant of Carleson's theorem on convergence of Fourier series, the development of quadratic Fourier analysis in the context of wave packet analysis and its various applications, and various convergence theorems for ergodic averages going beyond Bourgain's return times theorem. With its many applications to fields within mathematical analysis, the project helps to advance our understanding of the field in a variety of directions and thus has potential impact on several lines of research. In the past, analysis closely related to this project has found its way into applications to the non-mathematics world, most notably through the "wavelet" revolution in computational mathematics in the nineties. It is important to maintain our basic research in the field to lay out the groundwork for future applications. In addition, UCLA maintains an active group in analysis, training many young mathematicians in the area and successfully placing them in academia. This project helps continuing this contribution to the future mathematical workforce.