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.

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