February Fourier Talks 2009

Ubiquitous in the use of Fourier methods is the notion of a dual pair of variables, with the two sides related in a basis of sorts; perhaps a summation or a direct integral decomposition. Since processes arising in signal analysis often involve scaling similarity leaving gaps in the data, e.g., fractals, the result may not be linear, thus dictating modifications in the usual Fourier duality. We explore ways to make precise notions of scale and self-similarity, and we use it to unify a number of basis constructions for fractals, and to analyze their expansions (bases which are computationally efficient). Applications are given to both wavelets and fractals. We further show how to use processes from probability, random walks on branches, and their path-space measures in the study of associated convergence questions. Particular emphasis: Infinite products in the analysis of wavelets, signals, and fractals. We also use tools from signal/image processing and from operator theory.