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Illinois/Missouri Applied Harmonic Analysis Seminar


November 14, 2009

Notes from the Open Problem Session

Pete Casazza - University of Missouri
A quantitative notion of redundancy for finite frames
Abstract: We will improve the customary definition of redundancy by providing quantitative measures in its place, which we call UPPER AND LOWER REDUNDANCIES, that match better with an intuitive understanding of redundancy for finite frames. This motivates a carefully chosen list of desired properties for upper and lower redundancies. The means to achieve these properties is to consider the maximum and minimum of a redundancy function, which is interesting in itself. The redundancy function is defined on the sphere of the Hilbert space and measures the concentration of frame vectors around each point. A complete characterization of functions on the sphere which coincide with a redundancy function for some frame is given. The upper and lower redundancies obtained from this function are shown to satisfy all of the intuitively desirable properties. In addition, the range of values they assume is characterized.

Charles Chui - University of Missouri - St. Louis
A multi-level approach to context-preserving smooth function extension
Abstract: We introduce a multi-level interpolation (MLI) approach to the study of context-preserving smooth function extension on manifolds, with application to image inpainting. Solution of the Dirichlet problem relative to some Sturm- Liouville differential operator is used as the ground level of the MLI, and “wavelet details” in terms of certain appropriate mixed differential boundary data are filled in, according to the desirable number of MLI levels. An error formula, in terms of integral diffusion operators, with Green’s functions of the lagged anisotropic differential operators as diffusion kernels, is formulated and applied to derive the order of approximation.

Matthew Fickus - The Air Force Institute of Technology
Filter bank fusion frames
Abstract: A fusion frame is a sequence of orthogonal projection operators whose sum can be inverted in a numerically stable way. When properly designed, fusion frames can provide redundant encodings of signals which are optimally robust against certain types of noise and erasures. However, up to this point, few implementable constructions of such frames were known; we show how to construct them using oversampled filter banks. To be precise, we first provide polyphase matrix-based characterizations of filter bank fusion frames. We then use these characterizations to construct fusion frame versions of discrete wavelet and Gabor transforms, emphasizing those specific filters whose frequency responses are well-behaved.

Joe Lakey - New Mexico State University
Time- and band-limiting
Abstract: This talk will survey some of the classical and recent results concerning operators composed of a projection onto a compact set in time, followed by a projection onto a compact set in frequency. Such "time- and bandlimiting" operators were studied by Landau, Slepian, and Pollak in a series of papers published in the Bell Systems Tech. Journal in the early 1960s. Among other important results, Landau and Pollak gave an initial precise statement of the "folklore" observation that the dimension of the space of signals that are essentially timelimited to a given duration and bandlimited to a given frequency band is the time-bandwidth product. Other useful versions were proved by Slepian in the early 1970s and by Landau and Widom in 1980. Further progress on time- and bandlimiting has been intermittent, but genuine recent progress has been made in terms of numerical analysis, sampling theory, and extensions to multiband signals, all driven to some extent by potential applications in wireless communications. After providing an outline of the historical developments in the mathematical theory of time- and bandlimiting, some details of the sampling theory and the multiband setting will be given.

Azita Mayeli - SUNY Stony Brook
Spin wavelets on the sphere and their discretizations into frames
Abstract: Scalar valued wavelets on the sphere have been intensively used in cosmology, for statistical analysis of Cosmic Microwave Background (CMB) temperature maps. In addition to measuring the temperature fluctuations in the CMB with high sensitivity, the WMAP satellite has measured the polarization of the CMB, but with much lower sensitivity. Much more accurate data will be available soon, from the newly-launched Planck satellite. It is clear that efficient tools are required for the analysis of the polarization data. Localized tools are needed, because of contamination of the data in a large region, called the mask, by emissions from the Milky Way. Spin wavelets, introduced and constructed for the first time by D. Geller and D. Marinucci, can be applied for these purposes. In this talk we shall first review spin functions as well as spin wavelets. Then we state our main results on the discretization of spin wavelets into nearly tight frames. These frames enjoy the space-frequency localization property, which makes them a powerful tool for data analysis and for reconstruction. This is a joint work with Daryl Geller.