Calculate the eigenfunctions of the spherical-cap concentration problem.

## Usage

tapers, eigenvalues, taper_order = SHReturnTapers (theta0, lmax, [degrees])

## Returns

tapers : float, dimension (lmax+1, (lmax+1)**2)
The spherical harmonic coefficients of the (lmax+1)**2 localization windows. Each column contains the coefficients of a single window that possesses non-zero coefficients for the single angular order specified in taper_order. The first and last rows of each column correspond to spherical harmonic degrees 0 and lmax, respectively, and the columns are arranged from best to worst concentrated.
eigenvalues : float, dimension ((lmax+1)**2)
The concentration factors of the localization windows.
taper_order : integer, dimension ((lmax+1)**2)
The angular order of the non-zero spherical harmonic coefficients in each column of tapers.

## Parameters

theta0 : float
lmax : integer
The spherical harmonic bandwidth of the localization windows.
degrees : integer, optional, dimension (lmax+1), default = 1
List of degrees to use when computing the eigenfunctions. Only those degrees where degrees[l] is non-zero will be employed.

## Description

SHReturnTapers will calculate the eigenfunctions (i.e., localization windows) of the spherical-cap concentration problem. Each column of the matrix tapers contains the spherical harmonic coefficients of a single window and the corresponding concentration factor is given in the array eigenvalues. Each window has non-zero coefficients for only a single angular order that is specified in taper_order: all other spherical harmonic coefficients for a given window are identically zero. The columns of tapers are ordered from best to worst concentrated, and the first and last rows of each column correspond to spherical harmonic degrees 0 and lmax, respectively. The localization windows are normalized such that they have unit power. If the optional vector degrees is specified, then the eigenfunctions will be computed using only those degrees where degrees(l) is not zero.

When possible, the eigenfunctions are calculated using the kernel of Grunbaum et al. 1982 and the eigenvalues are then calculated by integration using the definition of the space-concentration problem. Use of the Grunbaum et al. kernel is prefered over the space-concentration kernel as the eigenfunctions of the later are unreliable when there are several eigenvalues identical (within machine precision) to either 1 or zero. If, the optional parameter degrees is specified, and at least one element is zero for degrees greater or equal to abs(m), then the eigenfunctions and eigenvalues will instead be computed directly using the space-concentration kernel.

## References

Grunbaum, 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, 1982.

Simons, F. J., F. A. Dahlen, and M. A. Wieczorek, Spatiospectral concentration on a sphere, SIAM Review, 48, 504-536, 2006.

Wieczorek, M. A. and F. J. Simons, Localized spectral analysis on the sphere, Geophys. J. Int., 162, 655-675, 2005.

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