Frame di Gabor

How to obtain from a signal a time-frequency representation? Conversely, how to reconstruct the original signal from his tfr? In this thesis we develop the subsection of time-frequency analysis known as Gabor analysis. The objects used by this discipline to reconstruct functions are the Gabor frames. After recalling some de?nitions and results about time-frequency analysis and frames we show theorems and propositions that constitute the base of modern Gabor analysis, proving crucial conditions for the boundedness of Gabor frame operator and deriving results about the existence of Gabor frames. We also prove some necessary conditions for Gabor frames and we meet two of the main properties of Gabor systems: the Wexler-Raz biorthogonality relations and the Ron-Shen duality principle. In the last chapter we leave the path of basic results and we present the state of the art of Gabor frames: we deal with modulation spaces and with the crucial notion of frame set, we generalize some previous results for non-rectangular lattices, and we illustrate a structural characterization for frame sets with windows in the Feichtinger's algebra. Then we focus on totally positive functions, that are, nowadays, the only kind of window for which are proved some results about the simplicity of the frame set. We conclude with a couple of theorems that ?x some obstructions for the existence of Gabor frames and we see what are the main open problems of Gabor analysis.