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computedm.doc
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computedm.doc
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Compute the space-concentration kernel of a spherical cap.
Usage
-----
dm = ComputeDM (lmax, m, theta0, [degrees])
Returns
-------
dm : float, dimension (lmax+1, lmax+1)
The space-concentration kernel or angular order m.
Parameters
----------
lmax : integer
The spherical harmonic bandwidth of the windows.
m : integer
The angular order of the concentration problem.
theta0 : float
The angular radius of the spherical cap in radians.
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.
Description
-----------
ComputeDM will calculate the space-concentration kernel of angular order m for
the spherical-cap concentration problem. The eigenfunctions of this matrix
correspond to a family of orthogonal windowing functions, and the eigenvalues
correspond to the window's concentration factor (i.e., the power of the window
within theta0 divided by the total power of the function). It is assumed that
the employed spherical harmonic functions are normalized to the same value for
all degrees and angular orders, which is the case for both the geodesy 4-pi and
orthonormalized harmonics. This kernel is symmetric and is computed exactly by
Gauss-Legendre quadrature. If the optional vector degrees is specified, then the
matrix will be computed only for elements where degrees(l) is not zero.
References
----------
Simons, F.J., F.A. Dahlen, and M.A. Wieczorek, Spatiospectral concentration on a
sphere, SIAM Review, 48, 504-536, 2006.