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Slepian Functions and Their Use in Signal Estimation and Spectral Analysis

  • Frederik J. Simons

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 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 on the surface of a sphere.

Key words:

Inverse theory Satellite geodesy Sparsity Spectral analysis Spherical harmonics Statistical methods 

Notes

Acknowledgment

For this work, I am most indebted to Tony Dahlen (1942–2007), for sharing my interest in this subject and being my friend, teacher, and mentor until his untimely death. I also thank Mark Wieczorek and Volker Michel for many enlightening discussions on this and similar topics over the last several years. Dong V. Wang aided with the computations for the Cartesian case. Financial support for this work was provided by the U. S. National Science Foundation under Grants EAR-0105387 and EAR-0710860, and by the Université Paris Diderot–Paris 7 and the Institut de Physique du Globe de Paris in St. Maur-des-Fossés, where this contribution was completed. Computer algorithms are made available on www.frederik.net.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Frederik J. Simons
    • 1
  1. 1.Department of GeosciencesPrinceton UniversityPrinceton, NJUSA

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