Calculate the eigenfunctions of the spherical-cap concentration problem for a single angular order.

## Usage

`tapers`

, `eigenvalues`

= SHReturnTapersM (`theta0`

, `lmax`

, `m`

, [`degrees`

])

## Returns

`tapers`

: float, dimension (`lmax`

+1,`lmax`

+1)- The spherical harmonic coefficients of the
`lmax+1`

localization windows, arranged in columns. 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)- The concentration factors of the localization windows.

## Parameters

`theta0`

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

: integer- The spherical harmonic bandwidth of the localization windows.
`m`

: integer- The angular order 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

`SHReturnTapersM`

will calculate the eigenfunctions of the spherical-cap concentration problem for a singule angular order. The 4pi normalized spherical harmonic coefficients of each window are given in the columns of `tapers`

, and the corresponding concentration factors are given in `eigenvaules`

. 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 `l`

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

shreturntapers, computedg82, computedm

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