TY - JOUR

T1 - Spatiospectral concentration of vector fields on a sphere

AU - Plattner, Alain

AU - Simons, Frederik Jozef

N1 - Funding Information:
A.P. and F.J.S. thank Volker Michel for valuable discussions, and the Ulrich Schmucker Memorial Trust and the Swiss National Science Foundation for financial support. F.J.S. thanks Tony Dahlen (1942–2007) and Liying Wei for their contributions to the material developed in this manuscript. We thank the anonymous referee and the Editor-in-Chief Charles Chui. This work was partially supported by National Science Foundation grants EAR-1014606 and EAR-1150145 to F.J.S. To the historians of science we like to mention that we discovered reference [79] only after concluding our analysis, when completing our manuscript for submission.

PY - 2014/1

Y1 - 2014/1

N2 - We construct spherical vector bases that are bandlimited and spatially concentrated, or, alternatively, spacelimited and spectrally concentrated, suitable for the analysis and representation of real-valued vector fields on the surface of the unit sphere, as arises in the natural and biomedical sciences, and engineering. Building on the original approach of Slepian, Landau, and Pollak we concentrate the energy of our function bases into arbitrarily shaped regions of interest on the sphere, and within certain bandlimits in the vector spherical-harmonic domain. As with the concentration problem for scalar functions on the sphere, which has been treated in detail elsewhere, a Slepian vector basis can be constructed by solving a finite-dimensional algebraic eigenvalue problem. The eigenvalue problem decouples into separate problems for the radial and tangential components. For regions with advanced symmetry such as polar caps, the spectral concentration kernel matrix is very easily calculated and block-diagonal, lending itself to efficient diagonalization. The number of spatiospectrally well-concentrated vector fields is well estimated by a Shannon number that only depends on the area of the target region and the maximal spherical-harmonic degree or bandwidth. The spherical Slepian vector basis is doubly orthogonal, both over the entire sphere and over the geographic target region. Like its scalar counterparts it should be a powerful tool in the inversion, approximation and extension of bandlimited fields on the sphere: vector fields such as gravity and magnetism in the earth and planetary sciences, or electromagnetic fields in optics, antenna theory and medical imaging.

AB - We construct spherical vector bases that are bandlimited and spatially concentrated, or, alternatively, spacelimited and spectrally concentrated, suitable for the analysis and representation of real-valued vector fields on the surface of the unit sphere, as arises in the natural and biomedical sciences, and engineering. Building on the original approach of Slepian, Landau, and Pollak we concentrate the energy of our function bases into arbitrarily shaped regions of interest on the sphere, and within certain bandlimits in the vector spherical-harmonic domain. As with the concentration problem for scalar functions on the sphere, which has been treated in detail elsewhere, a Slepian vector basis can be constructed by solving a finite-dimensional algebraic eigenvalue problem. The eigenvalue problem decouples into separate problems for the radial and tangential components. For regions with advanced symmetry such as polar caps, the spectral concentration kernel matrix is very easily calculated and block-diagonal, lending itself to efficient diagonalization. The number of spatiospectrally well-concentrated vector fields is well estimated by a Shannon number that only depends on the area of the target region and the maximal spherical-harmonic degree or bandwidth. The spherical Slepian vector basis is doubly orthogonal, both over the entire sphere and over the geographic target region. Like its scalar counterparts it should be a powerful tool in the inversion, approximation and extension of bandlimited fields on the sphere: vector fields such as gravity and magnetism in the earth and planetary sciences, or electromagnetic fields in optics, antenna theory and medical imaging.

KW - Bandlimited function

KW - Concentration problem

KW - Eigenvalue problem

KW - Spectral analysis

KW - Vector spherical harmonics

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U2 - 10.1016/j.acha.2012.12.001

DO - 10.1016/j.acha.2012.12.001

M3 - Article

AN - SCOPUS:84888308771

SN - 1063-5203

VL - 36

SP - 1

EP - 22

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

IS - 1

ER -