Mathematical Sciences: Problems in Wavelet and Sampling Theory

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Grochenig 9306430 This project is concerned mathematical problems arising in wavelet and sampling theory. Particular emphasis will be placed on the analysis of multidimensional problems. The thread which ties together the theories comes from the broader concept of frames developed in the early 1950's, but not recognized for its importance in new developments in signal processing. One objective of the work is the construction of orthogonal and biorthogonal wavelet bases for a general class of dilation matrices in higher dimensions. A particular step towards this construction is the characterization of all self-similar tilings which occur naturally in any attempt to find solutions ofdilation equations. The general construction of nonseparable wavelets will exploit the links with the theory of self-similar tilings. This type of wavelet basis is needed in applications to coding, compression and fast transmission images to obtain more flexibility with respect to the number of wavelets, the angular frequency distribution and the sampling geometry. The second objective is the development of algorithms for the reconstruction of band-limited multivariate functions from irregular samples. Particular emphasis will be given to the relation between finite-dimensional implementable models of irregular sampling to the original infinite-dimensional reconstruction problem, and th analysis of efficiency and speed of iterative reconstructions. The growth of wavelet analysis has transformed a sizable portion of research in harmonic analysis into efforts to find efficient and fast procedures for analyzing and transmitting signals in diverse areas such as tomography, seismology, data compression and image reconstruction. Many of the concepts have been known in various forms in the mathematical literature, but recent applications and discoveries have added impetus to the basic goal of local analysis of signals in both time and frequency at different scales in a comput ationally efficient manner. ***