# Prolates on the sphere, Extensions and Applications: Slepian functions for geophysical and cosmological signal estimation and spectral analysis

# Prolates on the sphere, Extensions and Applications: Slepian functions for geophysical and cosmological signal estimation and spectral analysis

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. In cosmology we may wish to compute the power spectral density of the cosmic microwave background radiation without the contaminating effect of the galactic plane.Analyzing and representing scientific data of this kind is facilitated in a basis of functions that are spatiospectrally concentrated, i.e. localized in both domains at the same time. Here, we give a theoretical overview of the approach to this concentration problem that was 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. "Spherical Slepian functions" are now widely applied to the study of inverse problems with (potential-field) satellite data, including such problems whose solutions are linear (source estimation), and quadratic (spectral estimation), in the data. Among the applications that I will discuss are the analysis of the time-variable gravity field for the recovery of coseismic gravity perturbations, the sparse analysis and representation of the lithospheric magnetic field, the recovery of the power spectral density of the cosmic microwave background radiation, and so on.