Compute the space-concentration kernel of an arbitrary mask on the sphere.

## Usage

`dij`

= ComputeDMap (`dh_mask`

, `lmax`

, [`n`

, `sampling`

, `degrees`

])

## Returns

`dij`

: float, dimension ( (`lmax`

+1)**2, (`lmax`

+1)**2 )- The space-concentration kernel corresponding to the mask dh_mask.

## Parameters

`dh_mask`

: integer, dimension (`nin`

,`sampling`

*`nin`

)- 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 maximum spherical harmonic degree of the matrix
`dij`

. `n`

: optional, integer, default =`nin`

- The number of latitudinal samples in
`dh_mask`

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

. `degrees`

: integer, optional, dimension (`lmax`

+1), default = 1- List of degrees to use when computing the space-concentration kernel. Only those degrees where
`degrees[l]`

is non-zero will be employed. `sampling`

: optional, integer, default determined by dimensions of`dh_mask`

- For 1,
`dh_mask`

has`n`

x`n`

samples. For 2,`dh_mask`

has`n`

x`2n`

samples.

## Description

`ComputeDMap`

will calculate the space-concentration kernel for a generic mask defined on the sphere. 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 `dij`

are calculated (see below), `sampling=2`

should be preferred. `dij`

is symmetric, and the elements are ordered according to the scheme described in `YilmIndexVector`

. See Simons et al. (2006) for further details. If the optional vector `degrees`

is specified, then the matrix will be computed only for elements where `degrees(l)`

is not zero.

The elements of DIJ are explicitly given by

`Dlm,l'm' = 1/(4pi) Integral_R Ylm Yl'm' dOmega`

,

where `R`

is the concentration region. In this routine, all values of `l'm'`

are calculated in a single spherical harmonic transform for a given value of `lm`

according to

`Dl'm' = 1/(4pi) Integral_Omega F Yl'm' dOmega`

.

where

`F = Ylm dh_mask`

.

The function `F`

is in general not a polynomial, and thus the coefficients `Dl'm'`

should not be expected to be exact. For this reason, the effective spherical harmonic degree of the input mask (`L=n/2-1`

) should be greater than `lmax`

. The exact value of `n`

should be chosen such that further increases in `n`

do not alter the returned eigenvalues. The routine prints out the fractional area of the mask computed in the pixel domain divided by `D(0,0)`

(the fractional area computed by the spherical harmonic transforms), and the ratio of the two should be close to 1. Experience suggests that `l`

should be about 5 times `lmax`

.

## 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

shreturntapersmap, yilmindexvector

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