Mathematical Sciences: Analysis of Multi-Dimensional Dilation Equations

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Scaling equations form a class of functional equations arising in a wide variety of applications including wavelets, sub-division schemes for computer generation of curves and surfaces and iterated function systems, the fundamental tools of fractal geometry. This project continues mathematical research on the representation of solutions to scaling equations (scaling functions) in terms of infinite matrix product expansions. At the present time, the convergence of the product is dependent on a contractivity assumption. The terms of the infinite product are chosen from a finite set of matrices and combined in a certain deterministic manner. That all possible finite products appear lead to investigations into bounded semi-groups of matrices. Work will be done investigating weaker assumptions involving bounded semi-groups and establishing weaker contractivity conditions on the corresponding products in terms of the Lyapunov exponent. A second goal of this project is that of using infinite matrix product expansions to establish iterated function system algorithms for the construction of scaling functions and to analyze their regularity and dimensionality. The work will concentrate primarily in the multi-dimensional setting where the most important applications lie. At the heart of wavelet theory is the simultaneous local decomposition of functions in space and frequency. Applications of this goal abound in mathematical analysis, signal processing, statistics and data compression. In combining the underlying principles with that of iterated function systems, this work leads naturally to important techniques for compression and reconstruction of television and other communication signals.