Calculate the expectation of the product of two functions, each multiplied by a different data taper, for a given spherical harmonic degree and two different angular orders.

## Usage

`value`

= SHSjkPG (`incspectra`

, `l`

, `m`

, `mprime`

, `hj_real`

, `hk_real`

, `mj`

, `mk`

, `lwin`

, `hkcc`

)

## Returns

`value`

: complex- The expectation of the product of two functions, each multiplied by a different data taper, for a given spherical harmonic degree and two different angular orders.

## Parameters

`incspectra`

: float, dimension (`l`

+`lwin`

+1)- The global cross-power spectrum of
`f`

and`g`

. `l`

: integer- The spherical harmonic degree for which to calculate the expectation.
`m`

: integer- The angular order of the first localized function,
`Phi`

. `mprime`

: integer- The angular order of the second localized function,
`Gamma`

. `hj_real`

: float, dimension (`lwin`

+1)- The real spherical harmonic coefficients of angular order
`mj`

used to localize the first function`f`

. These are obtained by a call to`SHReturnTapers`

. `hk_real`

: float, dimension (`lwin`

+1)- The real spherical harmonic coefficients of angular order
`mk`

used to localize the second function`g`

. These are obtained by a call to`SHReturnTapers`

. `mj`

: integer- The angular order of the window coefficients
`hj_real`

. `mk`

: integer- The angular order of the window coefficients
`hk_real`

. `lwin`

: integer- the spherical harmonic bandwidth of the localizing windows
`hj_real`

and`hk_real`

. `hkcc`

: integer- If 1, the function described in the
`description`

will be calculated as is. If 2, the second localized function`Gamma`

will not have its complex conjugate taken.

## Description

`SHSjkPG`

will calculate the expectation of two functions (`f`

and `g`

), each localized by a different data taper that is a solution of the spherical cap concentration problem, for a given spherical harmonic degree and two different angular orders. As described in Wieczorek and Simons (2007), this is the function

```
/ m(j) mprime(k)* \
| Phi Gamma |
\ l l /
```

The global cross-power spectrum of `f`

and `g`

is input as `incspectra`

, and the real coefficients of the two data tapers of angular order `mj`

and `mk`

(obtained by a call to `SHReturnTapers`

) are specified by `hj_real`

and `hk_real`

. If `hkcc`

is set to 1, then the above function is calculated as is. However, if this is set to 2, then the complex conjugate of the second localized function is not taken.

## References

Wieczorek, M. A. and F. J. Simons, Minimum-variance multitaper spectral estimation on the sphere, J. Fourier Anal. Appl., 13, doi:10.1007/s00041-006-6904-1, 665-692, 2007.