Compute the tridiagonal matrix of Grunbaum et al. (1982) that commutes with the space-concentration kernel of a spherical cap.

## Usage

dg82 = ComputeDG82 (lmax, m, theta0)

## Returns

dg82 : float, dimension (lmax-abs(m)+1, lmax-abs(m)+1)
The tridiagonal matrix of Grunbaum et al. (1982) that commutes with the space-concentration kernel of order M of a spherical cap.

## Parameters

lmax : integer
The spherical harmonic bandwidth of the windows.
m : integer
The angular order of the concentration problem.
theta0 : float

## Description

ComputeDG82 will calculate the tridiagonal matrix of Grunbaum et al. (1982) that commutes with the space-concentration kernel of order m of a spherical cap. 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. The returned matrix is symmetric, and the first element corresponds to (abs(m), abs(m)) as the values for elements less than this are identically zero.

## 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.