Given a window function which generates a Gabor (resp. Wavelet) frame. We consider the best approximation by those window functions that generate normalized tight (or just tight) frames. Using a parameterizations of window functions by certain class of operators in the von Neumann algebras associated with shift operators in time and frequency over certain lattices, we are able to prove that for any window function of a Gabor frame, there exists a unique window function which generates a tight Gabor frame and is the best approximation (among all the tight Gabor frames) for the given window function. More generally, we show that this is true for any frame induced by a projective unitary representation for a group. However, this result is not valid for wavelet frames. We will provide a restricted approximation result for semi-orthogonal wavelet frames.
We discuss three applications of operator algebra techniques in Gabor analysis: the parametrizations of Gabor frames, the incompleteness property, and the unique Gabor dual problem for subspace Gabor frames.
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