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Gabor systems and the Balian-Low Theorem

  • John J. Benedetto
  • Christopher Heil
  • David F. Walnut
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

The Balian-Low theorem (BLT) is a key result in time-frequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system {su2πimbt g(tna)} m,n∈ℤ with ab = 1 forms an orthonormal basis for L 2(ℝ) then The BLT was later extended from orthonormal bases to exact frames. This paper presents a tutorial on Gabor systems, the BLT, and related topics, such as the Zak transform and Wilson bases. Because of the fact that (g′)⋀(γ) = 2πiγĝ(γ), the role of differentiation in the proof of the BLT is examined carefully. We include the construction of a complete Gabor system of the form {e 2πibmt g(ta n )} such that {(a n ,b m )} has density strictly less than 1, and an Amalgam BLT that provides distinct restrictions on Gabor systems {e 2πimbt g(tna)} that form exact frames.

Keywords

Orthonormal Basis Tight Frame Dual 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 1998

Authors and Affiliations

  • John J. Benedetto
  • Christopher Heil
  • David F. Walnut

There are no affiliations available

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