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- The angular radius of the spherical cap in radians.
`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.

## See also

shreturntapersm, computedg82, computedm

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