2013/06/13 by Frederik J. Simons, Alain Plattner, Simons, Frederik J. +1
Computer Science · Mathematics · Physics and Astronomy · #Data Analysis #FOS: Physical sciences #Geochemistry and Geologic Mapping #Morphological variations and asymmetry #Statistical and numerical algorithms #Statistics and Probability (physics.data-an) #physics.data-an
paper · pdf · doi:10.48550/arxiv.1306.3184
Submitted to the 2nd Edition of the Handbook of Geomathematics, edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verlag. This is a slightly modified but expanded version of the paper arxiv:0909.5368 that appeared in the 1st Edition of the Handbook, when it was called: Slepian functions and their use in signal estimation and spectral analysis
arxiv created 2013/06/13 · openalex publication_date 2013/06/13 · arxiv updated 2013/06/14 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
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, for scalar and vectorial signals defined on the surface of a unit sphere.