Zak Transform Methods

  • Karlheinz Gröchenig
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

Any Gabor system G(g, α,β ) possesses a doubly periodic structure. In a technical sense, we have exploited this property in Chapters 6 and 7 through periodization tricks and the Poisson summation formula. However, the sequential application to the variables x and ω separately leads to an unnatural asymmetry, which was most pronounced in Walnut’s representation. The Zak transform offers a convenient tool for a more symmetric treatment of Gabor systems. Since the Zak transform is just a version of the Poisson summation formula, it is related to Fourier series and can be discretized easily. In engineering it has therefore become the preferred tool for the analysis of Gabor frames. In many important special cases, the invertibility and the spectrum of the frame operator can be characterized explicitly and clearly using the Zak transform.

Keywords

Orthonormal Basis Riesz Basis Tight Frame Gabor Frame Frame Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Karlheinz Gröchenig
    • 1
  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA

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