Fundamental problems are addressed in wavelet theory, non-uniform sampling, frames, and the theory of spectral-tile duality. These problems are inextricably interwoven by concept and technique. Operator theory provides the major unifying framework, combined with an integration of ideas from a diverse spectrum of mathematics including classical Fourier analysis, noncommutative harmonic analysis, representation theory, operator algebras, approximation theory, and signal processing. For example, the construction, implementation, and ensuing theory of single dyadic orthonormal wavelets in Euclidean space requires significant input from all of these disciplines as well as deep spectral-tile results.
There is intrinsic mathematical importance in the aforementioned problems, and the solutions to be formulated have broad and creative implications, both for mathematics and for applications in engineering and physics. The topics of this project have direct bearing on fast acquisition and motion problems in MRI, as well as in formulating algorithms for compression and noise reduction by means of proper cochlear modelling. There are further applications in quantum computing and image processing, and the development of non-uniform sampling strategies by this project play a role in state of the art A/D conversion methods used in multifunction RF systems. These interdisciplinary applications depending on modern mathematical analysis have educational implications in terms of cross-fertilization of ideas and research opportunities for graduate students.
