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A Banach space of test functions for Gabor analysis

  • Hans G. Feichtinger
  • Georg Zimmermann
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

We introduce the Banach space S 0L 2 which has a variety of properties making it a useful tool in Gabor analysis. S 0 can be characterized as the smallest time-frequency homogeneous Banach space of (continuous) functions. We also present other characterizations of S 0 turning it into a very flexible tool for Gabor analysis and allowing for simplifications of various proofs. A careful analysis of both the coefficient and the synthesis mapping in Gabor theory shows that an arbitrary window in S 0 not only is a Bessel atom with respect to arbitrary time-frequency lattices, but also yields boundedness between S 0 and e 1. On the other hand, we can study properties of general L 2-atoms since they induce mappings from S 0 to S0. This enables us to introduce a new, very natural concept of weak duality of Gabor atoms, applying also to the classical pair of the Gauss-function and its dual function determined by Bastiaans. Using the established results, we show a variety of properties that are desirable in applications, like the continuous dependence of the canonical dual window on the given Gabor window and on the lattice; continuity of thresholding and masking operators from signal processing; and an algorithm for the reconstruction of bandlimited functions from samples of the Gabor transform in a corresponding horizontal strip in the time-frequency plane. We also present an approximate Balian-Low Theorem stating that for close-to-critical lattices, the dual Gabor atoms progressively lose their time-frequency localization.

Keywords

Banach Space Dual Pair Riesz Basis Absolute Convergence Gabor Frame 
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

  • Hans G. Feichtinger
  • Georg Zimmermann

There are no affiliations available

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