(October 25) Ioannis Konstantinidis: TAIP - An Introduction - Our goal is to present some of the current mathematical challenges arising from the study of an interesting new device called TAIP (Thin film Analog Image Processor).
(November 8) Radu Balan: Gabor Approximations of Stochastic Processes - In this talk I'm going to present a Gabor analysis point of view of stationary stochastic processes. The first part of the talk is devoted to Gabor analysis on amalgam spaces. Here I'll be interested in necessary and sufficient conditions on the analysis/synthesis windows to define bounded operators on W(L^2, l^inf) and l^inf(Z^2). In the second part, I'll analyse the optimal approximation (in the mean square, a la Karhunen-Loeve, sense) of a given stationary process for a fixed rational redundancy. Using the Zak transform, the optimal windows turn out to be generically ill-localized, similar to the Balian-Low phenomenon.
(November 15) Dale Mugler: Discrete Hermite Functions and the Fractional Fourier Transform - This talk will compare two methods for generating a discrete set of Hermite functions. The continuous-time Hermite functions are well known as the solution of a particular differential equation, but they are also eigenfunctions of the Fourier transform as well as an orthogonal set of functions that can be useful in signal representations. For the discrete case, the methods compared in this generate the discrete Hermite functions as eigenvectors of a matrix that commutes with a Fourier matrix. One method involves a strictly tridiagonal commuting matrix while the other involves a nearly tridiagonal form. Each one can be related to a difference equation. A "centered" Fourier matrix has certain advantages to the generation of the eigenvectors. The discrete fractional Fourier transform (FRFT) is also a focus of this talk, as the FRFT may be computed using discrete Hermite functions. Background concerning the continuous time FRFT will be presented as well as applications of the discrete case.
(November 29) Wojtek Czaja: Boundedness of Pseudodifferential Operators on Modulation Spaces - We study boundedness properties of pseudodifferentail operators of Weyl calculus on modulation spaces introduced by Feichtinger. These spaces play a similar role in Gabor analysis as Besov spaces for wavelets. Recently, Grochenig and Heil have shown that the operators with symbols in $M_{\infty, 1}$ are bounded on all modulation spaces, thus proving the importance of modulation spaces in the theory of pseudodifferential operators. We further study the boudedness properties of pseudodifferential operators by introducing classes of symbols related to modulation spaces. Our approach may serve as an approximation to the results of Grochenig and Heil. We also show how it is related to the work of other authors.
(February 6, 2002) John Benedetto: MRI signal reconstruction by Fourier frames on interleaving spirals - Interleaving spirals are a natural setting for attaining fast MRI signal reconstruction. Using results of Beurling and Landau, as well as quantitative coverings of the spectral domain by translates of the polar set of the target disk space E, harmonics for Fourier frames F are constructed on interleaving spirals to reconstruct signals on E. Because of weak Fourier frame bound estimates by standard calculations, implementation is addressed in terms of finite frame approximants of F and convex symmetric polygonal approximants of E. The methods are extended to motion problems in MRI, and to finite tight frames on multidimensional spheres.
(February 14) Yang Wang: The Uniformity of Non-Uniform Gabor Bases - Although non-uniform (irregular) Gabor bases have attracted a lot of attention there hasn't been a single documented example of a compactly supported orthonormal Gabor basis that is actually non-uniform (irregular). We show that if the support of the generating function is an interval then the Gabor system must be regular. We also show that truly irregular Gabor bases exist in higher dimensions.
(February 27) David Walnut: Reconstruction from averages and systems of exponentials - We show that certain problems related to the local reconstruction of a function from its averages are related to the completeness and stability properties of certain systems of exponentials. These local reconstruction problems are related to deconvolution from multiple kernels and also to local Pompeiu problems.
(May 13) Carlos A. Cabrelli: Shift-invariant spaces revisited - In this talk we analyze the structure of shift-invariant spaces. We consider the case of a single compactly supported refinable generator. The spectral properties of a finite matrix provide useful information about the structure of the transition operator, and the assumption of global linear independence of the integer translates of the generator allows us to get results on the structure of the shift-invariant space itself. This information could be potentially useful in the construction of wavelets, sampling theory in shift invariant spaces, approximation theory, and many other topics. The extension of these results to finitely many generators and higher dimensions is in progress.
(May 13) Ursula M. Molter:
Hausdorff dimension of p-Cantor sets -
In this talk, we will discuss the construction of Cantor sets
associated to summable series of positive terms.
We will relate the Hausdorff dimension to the decay of the
sequence.