Gabor time-frequency lattices and the Wexler-Raz identity.

*(English)*Zbl 0888.47018Summary: 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 |