Spatiospectral concentration on a sphere

Frederik J. Simons, F. A. Dahlen, Mark A. Wieczorek

Research output: Contribution to journalReview article

221 Scopus citations

Abstract

We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere or, alternatively, of strictly spacelimited functions that are optimally concentrated in the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology, and numerical analysis. The spherical Slepian functions can be found by solving either an algebraic eigenvalue problem in the spectral domain or a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap, the spatiospectral projection operator commutes with a Sturm-Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small spatial region and a large spherical harmonic bandwidth, the spherical concentration problem reduces to its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.

Original languageEnglish (US)
Pages (from-to)504-536
Number of pages33
JournalSIAM Review
Volume48
Issue number3
DOIs
StatePublished - Sep 2006

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Theoretical Computer Science
  • Applied Mathematics

Keywords

  • Bandlimited function
  • Commuting differential operator
  • Concentration problem
  • Eigenvalue problem
  • Multitaper spectral analysis
  • Spherical harmonics

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