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A Guide to Concentration Alternating Frequency Response Analysis of Fuel Cells

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Published on: December 11, 2019

Eigenvalue estimates for Fourier concentration operators on two domains.

Felipe Marceca¹, José Luis Romero^2,3, Michael Speckbacher²

¹Department of Mathematics, King's College London, London, UK.

Archive for Rational Mechanics and Analysis

|April 15, 2024

Summary

This study introduces novel eigenvalue estimates for Fourier concentration operators, quantifying degrees of freedom for functions with spatial and frequency domain constraints. The findings offer precise, non-asymptotic bounds applicable to complex, non-convex domains.

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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.