Asymptotic Eigenvalue Distribution and Shannon Number for Slepian Spatiospectral Concentration on the d-dimensional Ball

Core Concepts

The eigenvalues of the Slepian spatiospectral concentration operator on the d-dimensional ball exhibit a bimodal distribution, clustering near 0 and 1. The number of significant non-zero eigenvalues (Shannon number) can be asymptotically characterized as the product of the bandwidth and a term depending on the spatial concentration domain.

Abstract

The content analyzes the asymptotic eigenvalue distribution and the Shannon number for the Slepian spatiospectral concentration problem on the d-dimensional unit ball Bd. Two different notions of bandwidth are considered:

Multivariate polynomials, where the bandlimit is defined by the maximal polynomial degree.
Fourier-Jacobi functions, where the bandlimit is defined by separate maximal degrees for the radial and spherical contributions.

For both setups, the authors prove that the eigenvalues of the corresponding composition operator exhibit a bimodal distribution, clustering near 0 and 1. Furthermore, they obtain asymptotic results for the number of significant non-zero eigenvalues (Shannon number), which can be expressed as the product of the bandwidth and a term depending on the spatial concentration domain.
The key results are:

For the multivariate polynomial setup, the weight function in the Shannon number formula is the Jacobi weight W0.
For the Fourier-Jacobi setup, the weight function is a modified version f
W0, which depends on the dimension d and the radial variable.

The proofs rely on the analysis of the trace and Hilbert-Schmidt norm of the composition operators, making use of results on the asymptotic behavior of reproducing kernels and orthogonal polynomials.

Stats

The number of significant non-zero eigenvalues (Shannon number) can be asymptotically characterized as:
♯{i : τ < λi(D; n) ≤1} ∼ N d
n ∫_D W0(x) dx
♯{i : τ < e
λi(D; m, n) ≤1} ∼ g
N d
mn ∫_D f
W0(x) dx
where N d
n and g
N d
mn are the dimensions of the underlying bandlimited function spaces, and W0 and f
W0 are the corresponding weight functions.

Quotes

"The eigenvalues of the spatiospectral concentration operators have a bimodal distribution, i.e., they cluster near zero and one."
"The number of significant non-zero eigenvalues (the so-called Shannon number) can be asymptotically characterized by the product of two quantities of which one solely depends on the bandwidth and the other one solely on the spatial concentration domain D ⊂Rd."

Key Insights Distilled From

Slepian spatiospectral concentration problem on the $d$-dimensional ball for different notions of bandwidth

by Christian Ge... at arxiv.org 04-24-2024

https://arxiv.org/pdf/2308.03543.pdf

Deeper Inquiries

How do the results extend to other notions of bandwidth beyond polynomials and Fourier-Jacobi functions

The results obtained for the Slepian spatiospectral concentration problem on the ball can be extended to other notions of bandwidth beyond polynomials and Fourier-Jacobi functions by considering different sets of basis functions that satisfy certain properties. For example, one could explore band-limiting operators defined using other orthogonal function systems such as Bessel functions, Laguerre functions, or even wavelet bases. By adapting the analysis to these alternative basis functions, one can investigate the eigenvalue distribution and concentration properties in different function spaces on the d-dimensional ball. The key is to ensure that the chosen basis functions form a complete and orthonormal system that allows for the decomposition of functions in the space of interest.

What are potential applications of the Slepian spatiospectral concentration problem on the ball in areas like geophysics, medical imaging, or other domains

The Slepian spatiospectral concentration problem on the ball has various potential applications in different fields such as geophysics, medical imaging, and signal processing. In geophysics, this problem can be utilized to analyze seismic data, gravitational modeling, or electromagnetic imaging within spherical domains. By studying the concentration of energy in spatial subdomains of the d-dimensional ball, geophysicists can better understand the distribution of geological structures or subsurface features. In medical imaging, the problem can aid in the localization of abnormalities or specific structures within spherical regions, improving the accuracy of diagnostic imaging techniques. Furthermore, in signal processing applications, the concentration of energy in specific frequency bands within spherical domains can enhance the efficiency of data compression, filtering, or feature extraction algorithms.

Can the asymptotic analysis be further refined to obtain sharper bounds on the eigenvalue distribution and Shannon number

The asymptotic analysis of the eigenvalue distribution and Shannon number in the Slepian spatiospectral concentration problem on the ball can be further refined to obtain sharper bounds by considering more intricate properties of the basis functions and the spatial domain. One approach to refining the analysis is to investigate the behavior of the eigenvalues near the boundaries of the spatial domain or at specific critical points within the domain. By studying the spectral properties of the composition operators in more detail, one can potentially derive tighter estimates on the number of significant eigenvalues and their distribution. Additionally, incorporating advanced mathematical techniques such as spectral graph theory or harmonic analysis may provide deeper insights into the concentration phenomenon and lead to more precise results on the eigenvalue distribution.