## Global spectral analysis

Function name Description
spectrum Calculate the spectrum of a real or complex function.
cross_spectrum Calculate the cross-spectrum of a real or complex function.
SHAdmitCorr Calculate the admittance and correlation spectra of two functions.
SHConfidence Compute the probability that two sets of spherical harmonic coefficients are correlated at a given degree and for a given correlation coefficient.

## Multitaper spectral estimation (spherical cap domain)

Function name Description
SHMultiTaperSE Perform a localized multitaper spectral analysis.
SHMultiTaperCSE Perform a localized multitaper cross-spectral analysis.
SHLocalizedAdmitCorr Calculate the localized admittance and correlation spectra of two functions at a given location.
SHReturnTapers Calculate the eigenfunctions of the spherical-cap concentration problem.
SHReturnTapersM Calculate the eigenfunctions of the spherical-cap concentration problem for a single angular order.
ComputeDm Compute the space-concentration kernel of a spherical cap.
ComputeDG82 Compute the tridiagonal matrix of Grünbaum et al. (1982) that commutes with the space-concentration kernel of a spherical cap.
SHFindLWin Determine the spherical-harmonic bandwidth that is necessary to achieve a certain concentration factor.
SHBiasK Calculate the multitaper (cross-)power spectrum expectation of a windowed function.
SHMTCouplingMatrix Calculate the multitaper coupling matrix for a given set of localization windows.
SHBiasAdmitCorr Calculate the expected multitaper admittance and correlation spectra associated with the input global cross-power spectra of two functions.
SHMTDebias Invert for the global power spectrum given a localized multitaper spectrum estimate.
SHMTVarOpt Calculate the minimum variance and corresponding optimal weights of a localized multitaper spectral estimate.
SHSjkPG Calculate the expectation of the product of two functions, each multiplied by a different data taper, for a given spherical harmonic degree and two different angular orders.
SHRotateTapers Rotate orthogonal spherical-cap Slepian functions centered at the North pole to a different location.

## Localization windows (arbitrary domain)

Function name Description
SHReturnTapersMap Calculate the eigenfunctions of the concentration problem for an arbitrary concentration region.
SHBiasKMask Calculate the multitaper (cross-)power spectrum expectation of a function localized by arbitrary windows derived from a mask.
SHMultiTaperMaskSE Perform a localized multitaper spectral analysis using arbitrary windows.
SHMultiTaperMaskCSE Perform a localized multitaper cross-spectral analysis using arbitrary windows.
ComputeDMap Compute the space-concentration kernel of a mask defined on the sphere.
Curve2Mask Given a set of latitude and longitude coordinates representing a closed curve, output a gridded mask.

## Localization bias (general)

Function name Description
SHBias Calculate the (cross-)power spectrum expectation of a windowed function.

## Slepian function expansions

Routine name Description
SlepianCoeffs Determine the expansion coefficients of a function for a given set of input Slepian functions.
SlepianCoeffsToSH Convert a function expressed in Slepian coefficients to spherical harmonic coefficients.

## Other routines

Function name Description
SphericalCapCoef Calculate the spherical harmonic coefficients of a spherical cap.

## References

• Grünbaum, F. A., L. Longhi, and M. Perlstadt, Differential operators commuting with finite convolution integral operators: some non-abelian examples, SIAM J. Appl. Math., 42, 941-955, doi:10.1137/0142067, 1982.