Wavelet frames, filters, and operator algebras

University Of Colorado At Boulder, Boulder CO

Investigators

Abstract

Packer will study wavelet frames and the related filter functions from the point of view of operator theory and operator algebras. Using this approach, Packer and her collaborators hope to shed light on both frame theory and filter functions and the related operators and operator algebras. She will attempt to construct more general families of wavelets, including wavelets based on fractal spaces, first introduced by D. Dutkay and P. Jorgensen, and on generalized solenoids. Operator theory and operator algebras were previously used by Packer with collaborators L. Baggett, K. Merrill, and Jorgensen, in their recent papers on the construction of Parseval frames from generalized filter systems. Packer intends to build on these results, and in addition hopes to use an approach due to N. Larsen and I. Raeburn, who have recently described a method to constructing multiresolution analyses by using directed systems of Hilbert spaces and partial isometries. The key linking ingredient in these projects is the family of filter functions, and these functions can also be used to construct operators giving representations of identifiable C*-algebras such as the Cuntz algebra and the Toeplitz algebra. Larsen, Raeburn, Merrill, Baggett and Packer intend to use this new approach to (1) build generalized frame wavelets from the abstract multiplicity function of Baggett and Merrill, and (2) give another approach towards constructing the fractal wavelets of Dutkay and Jorgensen. Packer also hopes to extend the concept of projective multiresolution analyses developed with M. Rieffel to more general non-commutative tori. This could have intriguing applications to Gabor frames. Packer, her student J. D'Andrea, and her colleague K. Merrill have recently used some of these methods on the space built from the Sierpinski gasket fractal. In so doing they were able to obtain some interesting analysis of digital photographs. Wavelet and frame theory are extremely useful in data storage and retrieval, both for signal processing and digital imaging. On the other hand, operator algebras play an important role in abstract mathematics and provide a bridge between analysis and geometry. The connection between the abstract theory of operator algebras and wavelets and their applications is an intriguing one, and it is well-known that Cuntz algebras and their generalizations have shown up in a variety of real-life situations over the years. Packer intends to build on her previous work with collaborators in the directions outlined above to obtain new results of interest to other workers in the field, thereby stimulating other research in this area. She hopes both to visit her domestic and overseas collaborators and have them visit Colorado, thereby benefiting a variety of institutions. Packer has had one student complete a Ph.D. thesis on this topic, and another of her students is interested in the use of fractal wavelets in image compression and data storage. A third student is interested in more theoretical problems in operator algebras. The mixture of the theoretical and the applied in this area of research generates many problems for graduate students to pursue, and Packer hopes to attract new Ph.D. students to work in these areas and related areas of analysis.