|Date:||Tue, March 16, 2010|
|Place:||Research I Seminar Room|
Abstract: In this talk, I will present an overview of my thesis, which addresses the following fundamental, but not yet fully understood questions from Gabor analysis. The first one is the construction of Gabor frames with smooth and compactly-supported windows for separable lattices in dimension greater than 2. Our approach is based on geometrical properties of fundamental domains of lattices and Fourier methods in translational tiling. Concrete examples are also provided to demonstrate the advantages and difficulties of smooth window design in higher dimensions. In addition, we review the interplay between harmonic analysis and representation theory. The theory of representations of operator algebras has been crucial in proving the general statement of the density theorem for Gabor frames, especially in the construction of multi-window Gabor frames for modulation spaces for arbitrary non-degenerate lattices in higher dimensions. The second question is the study of general classes of time-frequency localizing operators. We develop a general setup for identification based on an operator discretization method. Our approach is based on decay estimates for such localization operators, properties of Gaussian Gabor frames, and localization properties of Gabor molecules.