Mathematical Sciences: Harmonic Analysis and Self-Similarity

点击翻译按钮获取中文摘要

This project seeks to exploit some of the developing ideas in harmonic analysis which relate to important ideas in self-similarity. Self-similarity is a subject of intense interest today because of its relationship with fractal geometry and wavelet analysis. However, these relationships can be traced to work in the early 30's by Wiener and Wintner on Fourier transforms of Cantor measures. Work to be done includes the computation of dimensions of classes of self-similar measures. There are two directions in which the concept will be expanded. The first involves replacing strictly contractive transformations in their definition and replacing this with average contractivity. One can no longer guarantee compact support in this case. The growth rate of such measures and the smoothness of their Fourier transform are subjects for investigation. The second direction replaces the convex combinations with variable weights. Here one is interested in existence and uniqueness of the measures as well as their Lp and pointwise dimensions. Work will also be done in analyzing the Radon transform of self-similar measures. The two natural questions to be considered are whether or not the Radon transform is invertible and whether one can characterize the range of the transform. The transform is known to preserve the class of distributions if and only if the transform of the measure is not slowly decreasing. For self-similar measures this is not the case, leaving open the question of what the proper spaces ought to be. Harmonic analysis combines those elements of mathematics best exemplifying the ideas of synthesis. One seeks to decompose complex problems into fundamental components. These components are then analyzed for their basic characteristics. Finally, the solution is reconstructed through a recombination of the components. The Fourier series and Fourier transform are examples of tools used in this context; one discrete , the other representing a continuous decomposition. More recently the wavelet theory added new dimensions to some of the more classical approaches to harmonic analysis.