"Double Preconditioning for Gabor Frame Operators: Numerical and Functional Analytic Aspects"Balazs, PeterWe provide some numerical and analytic background, and arguments why and how the method of double preconditioning method introduced in 2006 allows to compute efficiently approximate dual (respectively tight) Gabor atoms for a given time-frequency lattice, and provides some new applications. We will extend the definition of the approach to the continuous setting,making use of the so-called Banach Gelfand Triple, based on Feichtinger's Algebra. We show the continuous dependency of the double preconditioning operators on their parameters. We also establish the connection to the POCS method for the solution of linear systems of equations. We discuss the influence of the order of the preconditioning approaches. In the applied section we show cases, when double preconditioning works very well. We adapt our method to find the canonical tight window and show that this allows to significantly generalize the FAB-method used in OFDM-applications. And finally, we use our approach as a fast, approximate way to generate dual windows for a slowly-varying lattice. |
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