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Minimal Frame Operator Norms Via Minimal Theta Functions.

Markus Faulhuber1

  • 1NuHAG, Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

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|May 15, 2018
PubMed
Summary

We found that hexagonal lattices minimize upper frame bounds for Gabor frames with standard Gaussian windows and even redundancy. This optimization uses advanced number theory results on minimal theta functions.

Keywords:
Frame boundsGabor framesTheta functions

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Area of Science:

  • Harmonic Analysis
  • Signal Processing
  • Mathematical Physics

Background:

  • Gabor frames are fundamental in signal processing for time-frequency analysis.
  • Understanding frame bounds is crucial for efficient and stable signal reconstruction.
  • Chirped Gaussian windows and various lattice structures are explored in frame theory.

Purpose of the Study:

  • To determine sharp frame bounds for Gabor frames using chirped Gaussians and rectangular lattices.
  • To investigate the equivalent case of standard Gaussian windows with general lattices.
  • To identify lattice structures that minimize upper frame bounds under specific conditions.

Main Methods:

  • Analysis of Gabor frames with chirped Gaussian windows and rectangular lattices.
  • Equivalence established to standard Gaussian windows and general lattices.
  • Application of Montgomery's result on minimal theta functions.

Main Results:

  • Sharp frame bounds are derived for the investigated Gabor frame configurations.
  • It is proven that for even redundancy and a standard Gaussian window, the hexagonal lattice minimizes the upper frame bound.
  • The study connects frame theory with number theory through theta functions.

Conclusions:

  • The hexagonal lattice offers optimal frame bounds for specific Gabor frame setups.
  • The findings contribute to the theoretical understanding of Gabor frame stability and efficiency.
  • This research provides insights into the geometric arrangements that optimize time-frequency representations.