Counting function estimates for coherent frames and Riesz sequences

Area of Science:

• Harmonic Analysis
• Functional Analysis
• Group Theory

Background:

• Coherent frames and Riesz sequences are fundamental structures in signal processing and mathematical analysis.
• Understanding their density properties is crucial for applications and theoretical development.
• Existing research provides foundational density conditions, but more precise estimates are needed.

Purpose of the Study:

• To derive asymptotic estimates for counting functions associated with coherent frames and Riesz sequences.
• To establish necessary density conditions for these structures on general unimodular amenable groups.
• To refine these estimates for systems with additional localization properties on groups of polynomial growth.

Main Methods:

• Asymptotic analysis of counting functions.
• Techniques from harmonic analysis and group theory.
• Exploration of localization properties within coherent systems.

Main Results:

• New estimates for the asymptotics of counting functions related to coherent frames and Riesz sequences.
• Recovery of known density conditions for these structures on unimodular amenable groups.
• More precise asymptotic estimates for localized coherent systems on groups of polynomial growth.

Conclusions:

• The study successfully establishes and refines density conditions for coherent frames and Riesz sequences.
• The findings offer improved analytical tools for understanding these mathematical objects.
• The results have implications for both the theoretical understanding and potential applications of frames and Riesz sequences.