Calculate the eigenfunctions and eigenvalues of the space-concentration problem for an arbitrary region.

## Usage

`tapers`

, `eigenvalues`

= SHReturnTapersMap (`dh_mask`

, `lmax`

, [`n`

, `ntapers`

, `sampling`

, `degrees`

])

## Returns

`tapers`

: float, dimension ((`lmax`

+1)**2,`ntapers`

)- The spherical harmonic coefficients of the tapers, arranged in columns, from best to worst concentrated. The spherical harmonic coefficients in each column are indexed according to the scheme described in
`YilmIndexVector`

. `eigenvalues`

: float, dimension (`ntapers`

)- The concentration factor for each localization window specified in the columns of
`tapers`

.

## Parameters

`dh_mask`

: integer, dimension (`n`

,`n`

*`sampling`

)- A Driscoll and Healy (1994) sampled grid describing the concentration region R. All elements should either be 1 (for inside the concentration region) or 0 (for outside R).
`lmax`

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

: optional, integer- The number of latitudinal samples in
`dh_mask`

. The effective spherical harmonic bandwidth of this grid is`L=n/2-1`

. `ntapers`

: optional, integer, default = (`lmax`

+1)**2- The number of best concentrated eigenvalues and corresponding eigenfunctions to return in
`tapers`

and`eigenvalues`

. The default value is to return all tapers. `sampling`

: optional, integer, default determined from`dh_mask`

- For 1,
`dh_mask`

has`n x n`

samples. For 2,`dh_mask`

has`n x 2n`

samples. The default it to determine`sampling`

from the dimensions of`dh_mask`

. `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

`SHReturnTapersMap`

will calculate the eigenfunctions (i.e., localization windows) of the space-concentration problem for an arbitrary concentration region specified in `dh_mask`

(see Simons et al. (2006) for further details). The input mask `dh_mask`

must be sampled according to the Driscoll and Healy (1994) sampling theorem with `n`

samples in latitude, and possess a value of 1 inside the concentration region, and 0 elsewhere. `dh_mask`

can either possess `n`

samples in longitude (`sampling=1`

) or `2n`

samples in longitude (`sampling=2`

). Given the approximate way in which the elements of the space-concentration kernel are calculated (see `ComputeDMap`

for details), `sampling=2`

should be preferred. The effective spherical harmonic bandwidth (L=N/2-1) of the grid `dh_mask`

determines the accuracy of the results, and experience shows that this should be about 3 times larger than `lmax`

.

The spherical harmonic coefficients of each window are given in the columns of `tapers`

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

. The spherical harmonic coefficients are ordered according to the scheme described in `YilmIndexVector`

, which can be converted to matrix form using `SHVectorToCilm`

, and the columns of `tapers`

are ordered from best to worst concentrated. The localization windows are normalized such that they have unit power. If the optional parameter `ntapers`

is specified, then only the `ntapers`

largest eigenvalues and corresponding eigenfunctions will be calculated and returned. If the optional vector `degrees`

is specified, then the eigenfunctions will be computed using only those degrees where `degrees(l)`

is not zero.

## References

Driscoll, J. R. and D. M. Healy, Computing Fourier transforms and convolutions on the 2-sphere, Adv. Appl. Math., 15, 202-250, 1994.

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

## See also

computedmap, yilmindexvector, shvectortocilm

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