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Gabor time-frequency lattices and the Wexler-Raz identity. (English) Zbl 0888.47018
Summary: Gabor time-frequency lattices are sets of functions of the form $$g_{m\alpha,n\beta}(t)= e^{-2\pi i\alpha mt}g(t-n\beta)$$ generated from a given function $$g(t)$$ by discrete translations in time and frequency. They are potential tools for the decomposition and handling of signals that, like speech or music, seem over short intervals to have well-defined frequencies that, however, change with time. It was recently observed that the behavior of a lattice $$(m\alpha,n\beta)$$ can be connected to that of a dual lattice $$(m/\beta, n/\alpha)$$. Here we establish this interesting relationship and study its properties. We then clarify the results of applying the theory of von Neumann algebras. One outcome is a simple proof that for $$g_{m\alpha,n\beta}$$ to $$\text{span }L^2$$, the lattice $$(m\alpha, n\beta)$$ must have at least unit density. Finally, we exploit the connection between the two lattices to construct expansions having improved convergence and localization properties.

##### MSC:
 47C15 Linear operators in $$C^*$$- or von Neumann algebras 94A11 Application of orthogonal and other special functions 46L05 General theory of $$C^*$$-algebras
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