Mathematical Sciences: Harmonic Analysis and Self-Similarity

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Abstract Strichartz 9623250 R. Strichartz will investigate problems related to self-similar measures, analysis on fractals, and an experimental approach to basic harmonic analysis estimates. The goal is to develop the theory of fractals using the tools of harmonic analysis, in the process enriching both areas. By making the central object a self-similar measure (a mass distribution that has similar structure at all scales) rather than a set, it becomes plausible to consider analytic questions, involving Fourier transforms, convolution operators, etc., as well as purely geometric questions. Analysis of functions defined on fractals (with respect to a self-similar measure) is an area of great interest in recent years. It is now possible to define the analogs of Fourier series and wavelet expansions for such functions. This project will extend this theory and answer some basic questions about these expansions. Self-similiarity is an idea that plays an important role in three areas of applied mathematics that have received a great deal of attention in recent years: wavelets, fractals, and quasicrystals. But the idea of self-similarity -- an object that can be broken up into pieces, each of which looks like a scaled down version of the whole -- can be investigated in a purely mathematical way that brings out its connections not only with these applications but also older areas of mathematics such as harmonic analysis. R. Strichartz has been in the forefront of the development of these ideas, and in this project will continue this line of investigation. The goal is to strengthen and enrich the mathematical infrastructure that makes possible the rapid development of important applications of mathematics.