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I have a set of solutions to an equation, which are all very similar to spherical harmonics. The solutions are discretised on a regular 3d grid. I would like to label them with which spherical harmonic they are most like, so the $l$ and $m$ values for each one.

Theoretically this is just selecting the largest coefficient from a decomposition of each solution vector onto the spherical harmonic space, or a transform of some kind.

Is there a cheap/simple way to calculate which harmonic each solution is nearest?

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When you say that a function on a regular three-dimensional grid is very similar to a spherical harmonic, do you mean that it's similar to the product of a radial function and a spherical harmonic? Since spherical harmonics are defined on the angular variables, a three-dimensional function can't literally be similar to a spherical harmonic. –  joriki Aug 1 '11 at 7:05
    
Well, keeping the radial function constant, i.e. f(r,theta,phi) ~ A.R.Y, so if R=R(r)=1, then f(x,y,z) ~ A.Y –  Phil H Aug 1 '11 at 7:21
    
A problem could arise if some solutions actually are linear combinations of spherical harmonics. In this case decomposition in the basis of spherical harmonics via scalar product could be of help here. –  Andrew Aug 1 '11 at 7:37

2 Answers 2

You could do a standard linear least-squares fit to a linear combination of spherical harmonics with variable coefficients:

$$f(\vec x)=\sum_{lm}c_{lm}Y_{lm}(\theta,\phi)$$

$$\sum_{\vec x}\left|f(\vec x)-f_{\vec x}\right|^2\rightarrow \min$$

$$\sum_{\vec x}\left|\sum_{lm}c_{lm}Y_{lm}(\theta,\phi)-f_{\vec x}\right|^2\rightarrow \min$$

$$\sum_{\vec x}Y^*_{lm}(\theta,\phi)\left(\sum_{l'm'}c_{l'm'}Y_{l'm'}(\theta,\phi)-f_{\vec x}\right)=0$$

$$\sum_{l'm'}\left(\sum_{\vec x}Y^*_{lm}(\theta,\phi)Y_{l'm'}(\theta,\phi)\right)c_{lm}=\sum_{\vec x}Y^*_{lm}(\theta,\phi)f_{\vec x}\;,$$

where $\theta$ and $\phi$ are the angular variables corresponding to $\vec x$, $f(\vec x)$ is the fitted function and $f_{\vec x}$ are the function values you have on the grid. This is a system of linear equations for the coefficients $c_{lm}$; you could solve it and say that your function is "nearest" to the spherical harmonic whose coefficient has the highest magnitude.

However, by using only function values within a spherical volume, you can consider the inner sum on the left-hand side as an approximation of an integral:

$$ \begin{eqnarray} \sum_{\vec x}Y^*_{lm}(\theta,\phi)Y_{l'm'}(\theta,\phi) &\approx& \int Y^*_{lm}(\theta,\phi)Y_{l'm'}(\theta,\phi)\mathrm dV \\ &=& \iint Y^*_{lm}(\theta,\phi)Y_{l'm'}(\theta,\phi)\mathrm d\Omega r^2\mathrm dr \\ &=& \delta_{ll'}\delta_{mm'}\int r^2\mathrm dr \\ &=& \frac13\delta_{ll'}\delta_{mm'}R^3\;, \end{eqnarray} $$

where $R$ is the radius of the spherical volume. This may actually be quite a good approximation if you have e.g. a cubical grid, since the symmetry of the grid causes many of the coefficients approximated as zero to actually vanish.

This radically simplifies the least-squares solution, since the matrix on the left is now the identity and the coefficients are simply given by the sums on the right.

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It really depends on what your functions are. Since you stated that your functions are very similar to spherical harmonics, I will assume that your functions are almost completely independent of the radius $r$ in spherical coordinates, and therefore it is well approximated by a function $f(\theta,\phi)$ defined over the unit sphere.

Now, if we also know, a priori, that the function is supposed to be very close to a fixed spherical harmonic, in the sense that we have a priori knowledge that the spherical harmonic decomposition

$$ f(\theta,\phi) = \sum c_{lm} Y_{lm}(\theta,\phi) $$

is such that there is a distinguished $l_0,m_0$ such that $c_{l_0m_0}$ is really large and $c_{l'm'}$ very small for $(l',m')\neq (l_0m_0)$ (and our goal is to find this pair $(l_0,m_0)$, you can make a guess at what $l_0$ is by considering that

$$ \int_{\mathbb{S}^2}|\nabla f|^2 dvol = -\int_{\mathbb{S}^2} f \triangle f \sim \int_{\mathbb{S}^2} (l_0+1)l_0|f|^2 dvol $$

So by considering

$$ \tilde{l} = \left\lfloor \sqrt{\frac{\int |\nabla f|^2}{\int f^2}}\right\rfloor $$

you get a good guess what $l_0$ should be. From this it requires to make a finite number of computations (compute the spherical harmonic coefficients for $l \in [\tilde{l}-2, \tilde{l}+2]$ and their accompanying admissible $m$ values, your need to just compute around $11l_0$ coefficients.

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