## 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.