Minimal Frame Operator Norms Via Minimal Theta Functions.

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.