Eigenvalue estimates for Fourier concentration operators on two domains.

Area of Science:

• Signal Processing
• Applied Mathematics
• Harmonic Analysis

Background:

• Concentration operators analyze functions supported on specific domains and their Fourier transforms on other domains.
• Understanding the spectral profile of these operators is crucial for determining prominent degrees of freedom in data analysis.
• Existing methods often struggle with non-convex or non-symmetric domains, limiting practical applications.

Purpose of the Study:

• To derive new, non-asymptotic eigenvalue estimates for Fourier concentration operators.
• To quantify the deviation of these operators from orthogonal projectors.
• To extend the analysis to non-convex and non-symmetric spatial and frequency domains.

Main Methods:

• Development of eigenvalue estimates based on the geometry of spatial and frequency domains.
• Utilizing redundant wave-packet expansions.
• Application of dyadic decomposition arguments for Schatten norm estimates of Hankel operators.

Main Results:

• Quantification of eigenvalues deviating from 0 and 1, providing bounds on degrees of freedom.
• Estimates are non-asymptotic, applicable to concrete domains and spectral thresholds.
• The study successfully addresses non-convex and non-symmetric domains, a novel contribution.

Conclusions:

• The derived estimates provide accurate, near-asymptotic benchmarks for Fourier concentration operators.
• This work expands the applicability of concentration operator theory to a wider range of real-world problems.
• The methods offer a robust framework for analyzing data with complex spatial and spectral support.