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Scalar and Vector Slepian Functions, Spherical Signal Estimation and Spectral Analysis

  • Frederik J.  Simons
  • Alain Plattner
Living reference work entry

Abstract

It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are “spatiospectrally” concentrated, i.e., “localized” in both domains at the same time. Here, we give a theoretical overview of one particular approach to this “concentration” problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions and particularly for applications in the geosciences and for scalar and vectorial signals defined on the surface of a unit sphere.

Keywords

Power Spectral Density Spherical Harmonic Coefficient Spherical Harmonic Degree Bandlimited Function Spherical Harmonic Basis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We are indebted to Tony Dahlen (1942–2007), Mark Wieczorek, and Volker Michel for many enlightening discussions over the years. Dong V. Wang aided with the calculations of the Cartesian case, and Liying Wei contributed to the development of the vectorial case. Yoel Shkolnisky pointed us to the symmetric relations (22) and (23), and Kornél Jahn shared a preprint of his most recent paper. Financial support for this work was provided by the US National Science Foundation under Grants EAR-0105387, EAR-0710860, EAR-1014606, and EAR-1150145, by the Université Paris Diderot–Paris 7 and the Institut de Physique du Globe de Paris in St. Maur-des-Fossés, the Ulrich Schmucker Memorial Trust, and the Swiss National Science Foundation. Computer algorithms are made available on www.frederik.net.

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Authors and Affiliations

  1. 1.Department of Geosciences, Princeton UniversityPrinceton, NJUSA
  2. 2.Program in Applied and Computational Mathematics, Princeton UniversityPrinceton, NJUSA
  3. 3.Department of Earth and Environmental Science, California State UniversityFresno, CAUSA

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