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computedg82.1
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computedg82.1
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.\" Automatically generated by Pandoc 1.17.2
.\"
.TH "computedg82" "1" "2016\-06\-17" "Fortran 95" "SHTOOLS 3.3"
.hy
.SH ComputeDG82
.PP
Compute the tridiagonal matrix of Grunbaum et al.
(1982) that commutes with the space\-concentration kernel of a spherical
cap.
.SH Usage
.PP
call ComputeDG82 (\f[C]dg82\f[], \f[C]lmax\f[], \f[C]m\f[],
\f[C]theta0\f[])
.SH Parameters
.TP
.B \f[C]dg82\f[] : output, real*8, dimension (\f[C]lmax\f[]\-abs(\f[C]m\f[])+1, \f[C]lmax\f[]\-abs(\f[C]m\f[])+1)
The tridiagonal matrix of Grunbaum et al.
(1982) that commutes with the space\-concentration kernel of order M of
a spherical cap.
.RS
.RE
.TP
.B \f[C]lmax\f[] : input, integer
The spherical harmonic bandwidth of the windows.
.RS
.RE
.TP
.B \f[C]m\f[] : input, integer
The angular order of the concentration problem.
.RS
.RE
.TP
.B \f[C]theta0\f[] : input, real*8
The angular radius of the spherical cap in radians.
.RS
.RE
.SH Description
.PP
\f[C]ComputeDG82\f[] will calculate the tridiagonal matrix of Grunbaum
et al.
(1982) that commutes with the space\-concentration kernel of order
\f[C]m\f[] of a spherical cap.
The eigenfunctions of this matrix correspond to a family of orthogonal
windowing functions, and the eigenvalues correspond to the window\[aq]s
concentration factor (i.e., the power of the window within
\f[C]theta0\f[] 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.
The returned matrix is symmetric, and the first element corresponds to
(abs(\f[C]m\f[]), abs(\f[C]m\f[])) as the values for elements less than
this are identically zero.
.SH References
.PP
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.
.PP
Simons, F.J., F.A.
Dahlen, and M.A.
Wieczorek, Spatiospectral concentration on a sphere, SIAM Review, 48,
504\-536, 2006.
.SH See also
.PP
computedm, shreturntapers, shreturntapersm