Theory and Applications of the Phi Transform Wavelets

摘要

A fundamental idea in Fourier analysis is that the Fourier Transform gives asimultaneous diagonalization of a small but very important class of operators including differentiation and integration. On the other hand, the Fourier Transform is not well suited for studying Multiplication operators. The wavelet transform (and related transforms) give excellent simultaneous almost diagonalization of a very large class of operators which includes differentiation, integration, and multiplication: in fact, more-generally singular integral operators and pseudo-differential operators. Professor Rochberg's recent work has been to use this fact to study such operators. Some work has been in the real variable tradition, other parts have involved operators on spaces of analytic functions.