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shreturntapers.doc
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shreturntapers.doc
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Calculate the eigenfunctions of the spherical-cap concentration problem.
Usage
-----
tapers, eigenvalues, taper_order = SHReturnTapers (theta0, lmax, [degrees])
Returns
-------
tapers : float, dimension (lmax+1, (lmax+1)**2)
The spherical harmonic coefficients of the (lmax+1)**2 localization windows.
Each column contains the coefficients of a single window that possesses non-
zero coefficients for the single angular order specified in taper_order. The
first and last rows of each column correspond to spherical harmonic degrees
0 and lmax, respectively, and the columns are arranged from best to worst
concentrated.
eigenvalues : float, dimension ((lmax+1)**2)
The concentration factors of the localization windows.
taper_order : integer, dimension ((lmax+1)**2)
The angular order of the non-zero spherical harmonic coefficients in each
column of tapers.
Parameters
----------
theta0 : float
The angular radius of the spherical cap in radians.
lmax : integer
The spherical harmonic bandwidth of the localization windows.
degrees : integer, optional, dimension (lmax+1), default = 1
List of degrees to use when computing the eigenfunctions. Only those degrees
where degrees[l] is non-zero will be employed.
Description
-----------
SHReturnTapers will calculate the eigenfunctions (i.e., localization windows) of
the spherical-cap concentration problem. Each column of the matrix tapers
contains the spherical harmonic coefficients of a single window and the
corresponding concentration factor is given in the array eigenvalues. Each
window has non-zero coefficients for only a single angular order that is
specified in taper_order: all other spherical harmonic coefficients for a given
window are identically zero. The columns of tapers are ordered from best to
worst concentrated, and the first and last rows of each column correspond to
spherical harmonic degrees 0 and lmax, respectively. The localization windows
are normalized such that they have unit power. If the optional vector degrees is
specified, then the eigenfunctions will be computed using only those degrees
where degrees(l) is not zero.
When possible, the eigenfunctions are calculated using the kernel of Grunbaum et
al. 1982 and the eigenvalues are then calculated by integration using the
definition of the space-concentration problem. Use of the Grunbaum et al. kernel
is prefered over the space-concentration kernel as the eigenfunctions of the
later are unreliable when there are several eigenvalues identical (within
machine precision) to either 1 or zero. If, the optional parameter degrees is
specified, and at least one element is zero for degrees greater or equal to
abs(m), then the eigenfunctions and eigenvalues will instead be computed
directly using the space-concentration kernel.
References
----------
Grunbaum, F. A., L. Longhi, and M. Perlstadt, Differential operators commuting
with finite convolution integral operators: Some non-abelian examples, SIAM, J.
Appl. Math. 42, 941-955, 1982.
Simons, F. J., F. A. Dahlen, and M. A. Wieczorek, Spatiospectral concentration
on a sphere, SIAM Review, 48, 504-536, 2006.
Wieczorek, M. A. and F. J. Simons, Localized spectral analysis on the sphere,
Geophys. J. Int., 162, 655-675, 2005.