Fourier transform
Spherical harmonics

Let the set of homogeneous harmonic polynomials of degree k on R be denoted by A. The set A consists of the solid spherical harmonics of degree k. The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, if f (x) = e P (x) for some P (x) in A, then f̂ (ξ) = i f (ξ). Let the set H be the closure in L (R) of linear combinations of functions of the form f (| x |) P (x) where P (x) is in A. The space L (R) is then a direct sum of the spaces H and the Fourier transform maps each space H to itself and is possible to characterize the action of the Fourier transform on each space H.

Let f (x) = f (| x |) P (x) (with P (x) in A), then f ^ (ξ) = F 0 (| ξ |) P (ξ) {\displaystyle {\hat {f}}(\xi)=F_{0}(|\xi |)P(\xi)} where F 0 (r) = 2 π i − k r − n + 2 k − 2 2 ∫ 0 ∞ f 0 (s) J n + 2 k − 2 2 (2 π r s) s n + 2 k 2 d s. {\displaystyle F_{0}(r)=2\pi i^{-k}r^{-{\frac {n+2k-2}{2}}}\int _{0}^{\infty }f_{0}(s)J_{\frac {n+2k-2}{2}}(2\pi rs)s^{\frac {n+2k}{2}}\,ds.}

Here J denotes the Bessel function of the first kind with order n + 2 k − 2 / 2. When k = 0 this gives a useful formula for the Fourier transform of a radial function. This is essentially the Hankel transform. Moreover, there is a simple recursion relating the cases n + 2 and n allowing to compute, e.g., the three-dimensional Fourier transform of a radial function from the one-dimensional one.
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