Expand an equally sampled or equally spaced complex grid into complex spherical harmonics using Driscoll and Healy’s (1994) sampling theorem.

## Usage

cilm = SHExpandDHC (griddh, [norm, sampling, csphase, lmax_calc])

## Returns

- cilm : complex, dimension (2, n/2, n/2) or (2, lmax_calc+1, lmax_calc+1)
- The complex spherical harmonic coefficients of the function. These will be exact if the function is bandlimited to degree lmax=n/2-1. The first index specifies the coefficient corresponding to the positive and negative order of m, respectively, with Clm=cilm[0,l,m] and Cl,-m=cilm[1,l,m].

## Parameters

- griddh : complex, dimension (n, n) or (n, 2*n)
- A 2D equally sampled (default) or equally spaced complex grid that conforms to the sampling theorem of Driscoll and Healy (1994). The first latitudinal band corresponds to 90 N, the latitudinal band for 90 S is not included, and the latitudinal sampling interval is 180/n degrees. The first longitudinal band is 0 E, the longitude band for 360 E is not included, and the longitudinal sampling interval is 360/n for an equally and 180/n for an equally spaced grid, respectively.
- norm : optional, integer, default = 1
- 1 (default) = 4-pi (geodesy) normalized harmonics; 2 = Schmidt semi-normalized harmonics; 3 = unnormalized harmonics; 4 = orthonormal harmonics.
- sampling : optional, integer, default = 1
- If 1 (default) the input grid is equally sampled (n by n). If 2, the grid is equally spaced (n by 2n).
- csphase : optional, integer, default = 1
- 1 (default) = do not apply the Condon-Shortley phase factor to the associated Legendre functions; -1 = append the Condon-Shortley phase factor of (-1)^m to the associated Legendre functions.
- lmax_calc : optional, integer, default = n/2-1
- The maximum spherical harmonic degree calculated in the spherical harmonic expansion.

## Description

SHExpandDHC will expand an equally sampled (n by n) or equally spaced complex grid (n by 2n) into complex spherical harmonics using the sampling theorem of Driscoll and Healy (1994). The number of latitudinal samples n must be even, and the transform is exact if the function is bandlimited to spherical harmonic degree n/2 - 1. The inverse transform is given by the routine MakeGridDHC. If the optional parameter lmax_calc is specified, the spherical harmonic coefficients will only be calculated to this degree instead of n/2 - 1. The algorithm is based on performing FFTs in longitude and then integrating over latitude using an exact quadrature rule.

The default is to use an input grid that is equally sampled (n by n), but this can be changed to use an equally spaced grid (n by 2n) by the optional argument sampling. When using an equally spaced grid, the Fourier components corresponding to degrees greater than n/2 - 1 are simply discarded; this is done to prevent aliasing that would occur if an equally sampled grid was constructed from an equally spaced grid by discarding every other column of the input grid.

The employed spherical harmonic normalization and Condon-Shortley phase convention can be set by the optional arguments norm and csphase; if not set, the default is to use geodesy 4-pi normalized harmonics that exclude the Condon-Shortley phase of (-1)^m. The normalized legendre functions are calculated in this routine using the scaling algorithm of Holmes and Featherstone (2002), which are accurate to about degree 2800. The unnormalized functions are accurate only to about degree 15.

## References

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

Holmes, S. A., and W. E. Featherstone, A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions, J. Geodesy, 76, 279-299, 2002.

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