shsjkpg.doc

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.