Slepian functions and their use in signal estimation and spectral analysis

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 on the surface of a sphere.

Submitted to the Handbook of Geomathematics, edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verlag

References in corpus (7)

• Minimum bias multiple taper spectral estimation
• Spatiospectral concentration on a sphere
• Spherical Slepian functions and the polar gap in geodesy
• Minimum-variance multitaper spectral estimation on the sphere
• Spectral estimation on a sphere in geophysics and cosmology
• Efficient analysis and representation of geophysical processes using localized spherical basis functions
• A spatiospectral localization approach to estimating potential fields on the surface of a sphere from noisy, incomplete data taken at satellite altitudes

Cited by in corpus (7)

• Spatiospectral concentration in the Cartesian plane
• Yield--Optimized Superoscillations
• Harmonic Singular Integrals and Steerable Wavelets in $L_2({\mathbb{R}}^d)$
• A General Approach to Regularizing Inverse Problems with Regional Data using Slepian Wavelets
• Efficient Representations of Signals in Nonlinear Signal Processing with Applications to Inverse Problems
• Spatiospectral concentration of vector fields on a sphere
• Quantum phase estimation with optimal confidence interval using three control qubits