Analyzing a signal in regard to its time-varying frequency content is a classical method in science. Recently this method has grown into a rich mathematical theory within applied harmonic analysis, with applications ranging from radio and mobile communications to medical image processing and geophysics. This thesis addresses the following three fundamental, but not yet fully understood questions. The first one is the construction of Gabor frames with smooth and compactlysupported window functions defined on the Euclidean plane for separable lattices. In addition, we review the crucial applications of the theory of operator algebra representations for proving the general statement of the density theorem for Gabor frames. Second, we study identification of incompletely known linear operators based on the observation of restricted input and output signals. We develop a general setup for identification of general classes of time-frequency localizing operators based on a discretization method. The third question is the uncertainty principle for the joint time-frequency representation of functions on finite Abelian groups. Algebraic properties of such groups lead to results relating the support sizes of functions and their short-time Fourier transforms, with applications in the construction of a class of equal norm tight Gabor frames that are maximally robust to erasures, and consequences in the theory of recovering and storing signals with sparse time-frequency representations.
['Peter Rashkov']
Time-frequency localized functions and operators in Gabor analysis