Editing Fourier transform (section)

=== Spherical harmonics ===
Let the set of [[Homogeneous polynomial|homogeneous]] [[Harmonic function|harmonic]] [[polynomial]]s of degree {{mvar|k}} on {{math|'''R'''<sup>''n''</sup>}} be denoted by {{math|'''A'''<sub>''k''</sub>}}. The set {{math|'''A'''<sub>''k''</sub>}} consists of the [[solid spherical harmonics]] of degree {{mvar|k}}. The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, if {{math|1=''f''(''x'') = ''e''<sup>−π{{abs|''x''}}<sup>2</sup></sup>''P''(''x'')}} for some {{math|''P''(''x'')}} in {{math|'''A'''<sub>''k''</sub>}}, then {{math|1=''f̂''(''ξ'') = ''i''{{isup|−''k''}} ''f''(''ξ'')}}. Let the set {{math|'''H'''<sub>''k''</sub>}} be the closure in {{math|''L''<sup>2</sup>('''R'''<sup>''n''</sup>)}} of linear combinations of functions of the form {{math|''f''({{abs|''x''}})''P''(''x'')}} where {{math|''P''(''x'')}} is in {{math|'''A'''<sub>''k''</sub>}}. The space {{math|''L''<sup>2</sup>('''R'''<sup>''n''</sup>)}} is then a direct sum of the spaces {{math|'''H'''<sub>''k''</sub>}} and the Fourier transform maps each space {{math|'''H'''<sub>''k''</sub>}} to itself and is possible to characterize the action of the Fourier transform on each space {{math|'''H'''<sub>''k''</sub>}}.<ref name="Stein-Weiss-1971" />

Let {{math|1=''f''(''x'') = ''f''<sub>0</sub>({{abs|''x''}})''P''(''x'')}} (with {{math|''P''(''x'')}} in {{math|'''A'''<sub>''k''</sub>}}), then
<math display="block">\hat{f}(\xi)=F_0(|\xi|)P(\xi)</math>
where
<math display="block">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.</math>

Here {{math|''J''<sub>(''n'' + 2''k'' − 2)/2</sub>}} denotes the [[Bessel function]] of the first kind with order {{math|{{sfrac|''n'' + 2''k'' − 2|2}}}}. When {{math|''k'' {{=}} 0}} this gives a useful formula for the Fourier transform of a radial function.<ref>{{harvnb|Grafakos|2004}}</ref> This is essentially the [[Hankel transform]]. Moreover, there is a simple recursion relating the cases {{math|''n'' + 2}} and {{mvar|n}}<ref>{{harvnb|Grafakos|Teschl|2013}}</ref> allowing to compute, e.g., the three-dimensional Fourier transform of a radial function from the one-dimensional one.