Celebratio Mathematica — Daubechies

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I. Daubechies, H. J. Landau, and Z. Landau: “Gabor time-frequency lattices and the Wexler–Raz identity,” J. Fourier Anal. Appl. 1 : 4 (1995), pp. 437–478. MR 1350701 Zbl 0888.47018 article

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 by applying the theory of von Neumann algebras. One outcome is a simple proof that for \( g_{m\alpha,n\beta} \) to 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.

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A. R. Calderbank and I. Daubechies: “The pros and cons of democracy,” pp. 1721–1725 in Shannon theory: Perspective, trends, and applications: Special issue dedicated to Aaron D. Wyner, published as IEEE Trans. Inform. Theory 48 : 6. Issue edited by H. J. Landau, J. E. Mazo, S. Shamai, and J. Ziv. IEEE (New York), June 2002. MR 1902984 Zbl 1061.94014 incollection

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