Assume that $f(x,y,z)$ is a function defined over the unit 2-sphere in three dimensions. Our aim is to explore the building blocks of $f$ using the partition command. Let's start with a spherefun object:
f = spherefun(@(x,y,z) 0.5 + sinh(5*x.*y.*z).*cos(x-y+2*z)) plot(f), axis off, hold on contour(f, 'color','k'),
f = spherefun object domain rank vertical scale unit sphere 21 1.6
A spherefun can be seen as a sum of two spherefuns, one of them even/$\pi$-periodic and the other odd/$\pi$-anti-periodic [1]. Recall that a univariate function $g$ is $\pi$-anti-periodic if $g(x+\pi) = -g(x)$. The command `[fep, foa] = partition(f)' partitions $f$ accordingly.
[fep, foa] = partition(f) err = norm(fep+foa - f) subplot(1,2,1), plot(fep), hold on, contour(fep,'k') title('even/periodic part'), axis off subplot(1,2,2), plot(foa), hold on, contour(foa,'k') title('odd/anti-periodic part'), axis off, axis off, hold off
fep = spherefun object domain rank vertical scale unit sphere 11 1.2 foa = spherefun object domain rank vertical scale unit sphere 10 0.94 err = 0
fep has a CDR decomposition [1] whose columns are even and whose rows are $\pi$-periodic (not just $2\pi$!):
[Ce, D, Rp] = cdr(fep); clf, plot(Ce) grid on, title('Columns of the even part of f')
clf, plot(Rp) grid on, title('Rows of the \pi-periodic part of f')
The other part of $f$, foa, has a CDR decomposition whose columns are odd and whose rows are $\pi$-anti-periodic:
[Co, D, Ra] = cdr(foa); plot(Co), grid on, title('Columns of the odd part of f')
clf, plot(Ra) grid on, title('Rows of the \pi-anti-periodic part of f')
The integral of a spherefun is equal to the integral of its even/$\pi$-periodic piece, since the integral of any odd/$\pi$-anti-periodic spherefun is zero:
format long sum_f = sum2(f) sum_foa = sum2(foa) sum_fep = sum2(fep)
sum_f = 6.283185307179586 sum_foa = 0 sum_fep = 6.283185307179586
An equivalent partitioning is available for diskfuns [2].
References
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A. Townsend, H. Wilber, and G. Wright, Computing with functions in spherical and polar geometries I. The sphere. SIAM J. Sci. Comput., 38 (2016) C403-C425.
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A. Townsend, H. Wilber, and G. Wright, Computing with functions in spherical and polar geometries II. The disk, Submitted (2016).