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What is the definition (or some sources) of the Fourier transform of spherical harmonics?

Ooker
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Mack
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    The spherical harmonics are functions with domain the sphere $S^2$. The standard embedding of the sphere in $\mathbb{R}^3$ does not have a group structure, which means the usual Fourier transform cannot be defined. Mathematically, this is one justification for introducing the spherical harmonics as a basis for $L^2(S^2)$... we don't have any characters. http://mathoverflow.net/questions/32800/spherical-harmonics-a-bunch-of-questions-about-them – dls Jul 11 '13 at 01:27
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    Look at the answers you received to your previous question. Also, check out Axler's book Harmonic Function Theory: http://www.axler.net/HFT.html (freely available PDF). – dls Jul 11 '13 at 01:29
  • Can you clarify whether you mean the polynomials (which are defined on all of $\mathbb R^n$), and genuine Euclidean Fourier transform, or do you mean the harmonic analysis on the sphere itself (which is about the representation theory of $O(n)$, a non-abelian, but compact, group). In the former case, those Fourier transforms can be nicely evaluated as Fourier transforms of tempered distributions, again using a little repn theory of $O(n)$. In the latter case, the spherical harmonics are themselves already the eigenfunctions in terms of which other things are expanded. – paul garrett Jul 17 '13 at 18:39

1 Answers1

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This problem can be done simply at least formally: use the plane wave expansion into spherical harmonics and spherical Bessel functions,

$$e^{i\vec{k}\cdot\vec{r}} = (4\pi)\sum_{lm} i^l j_l(kr) Y_{lm}(\hat{k})Y_{lm}^*(\hat{r})$$

We then have

$$FT\{Y_{l'm'}(\hat{r})\}(\hat{k})=\sum_{lm}Y_{lm}(\hat{k})i^l\int d^3\vec{r} j_l(kr) Y_{lm}^*(\hat{r})Y_{l'm'}(\hat{r}).$$

The angular integral can be done by orthogonality of the spherical harmonics, setting $lm = l'm'$, so we find

$$FT\{Y_{l'm'}(\hat{r})\}(\hat{k})=Y_{lm}(\hat{k})4\pi i^l \int r^2 dr j_l(kr)$$.

The integral of $j_l$ against $r^2 dr$ is divergent because as $x\to \infty$ all the spherical Bessel functions behave as $1/x$. Nonetheless this is a useful representation because one can then resolve the $j_l$ integral into a Dirac delta function or derivatives of it. For instance, using the identity

$$\int j_l(ax) j_l(bx) x^2 dx =\frac{1}{2\pi b^2}\delta_D(a-b)$$

with $l=0$ and $a=0$, the first spherical Bessel function goes to unity and we have the desired integral for $l=0$. One should be able to use recursion relations to write the higher order $j_l$ in terms of lower order $j_l$ and their derivatives, and also Rayleigh's formula to write integrals using parametric differentiation where needed.

For instance,

\begin{align} \int x^2 dx j_2(kx) &= \int x^2 dx j_0(kx) +\frac{3}{k} \int x dx j_1(kx)\\ &=\frac{\pi}{2k^2}\delta_D(k)+\frac{3\pi}{2k^3}. \end{align}

We used the relation $j_{n+1}(x)=(2n+1)j_n(x)/x-j_{n-1}(x)$ to split the integral and then for the second integral that $\int x j_1(kx) dx=-\partial/\partial k \int j_0(kx)dx$, an application of Rayleigh's formula.

Gabriel Sandoval
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zs_int
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