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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $f \\in L^1(\\R^n)$ be radial, that is, $f(x) = f_0(\\abs{x})$ for a measurable function $f_0$ on $(0, \\infty)$ with $\\int_0^\\infty \\abs{f_0(r)}\\, r^{n-1}\\, dr \u003c \\infty$.\nLet $\\hat f(\\xi) = \\int_{\\R^n} f(x) e^{-2\\pi i x \\cdot \\xi}\\, dx$ be its [[Fourier transform|DPVM34ADZV]], and let $J_\\nu$ denote the Bessel function of the first kind of order $\\nu$.\nThen $\\hat f$ is radial, and for every $\\xi \\ne 0$, with $\\rho = \\abs{\\xi}$,\n$$\\hat f(\\xi) = 2\\pi\\, \\rho^{1 - n/2} \\int_0^\\infty f_0(r)\\, J_{n/2 - 1}(2\\pi r \\rho)\\, r^{n/2}\\, dr.$$\n\n## Remarks\n\nThe transform commutes with [[rotations|0HFREYRSZP]]: if $R$ is an orthogonal matrix, then the transform of $f \\circ R$ is $\\hat f \\circ R$.\nSo the transform of a radial function is radial.\nTo compute it, write the integral in polar coordinates $x = r \\omega$ and use the formula $\\int_{S^{n-1}} e^{-2\\pi i r \\rho\\, \\omega \\cdot e_1}\\, d\\sigma(\\omega) = 2\\pi (r\\rho)^{1 - n/2} J_{n/2-1}(2\\pi r \\rho)$ for the surface measure $\\sigma$ on the unit sphere.\nThe integral converges absolutely, because $z^{-\\nu} J_\\nu(z)$ is bounded on $(0, \\infty)$ for $\\nu \\ge -\\frac12$ and the hypothesis on $f_0$ holds.\n\nExample: for $n = 3$, $J_{1/2}(z) = \\sqrt{2/(\\pi z)}\\, \\sin z$, and the formula becomes\n$$\\hat f(\\xi) = \\frac{2}{\\rho} \\int_0^\\infty f_0(r)\\, r \\sin(2\\pi r \\rho)\\, dr.$$\nFor $n = 1$ the same formula, with $J_{-1/2}(z) = \\sqrt{2/(\\pi z)} \\cos z$, reduces to the cosine transform of an even function.\n\nUp to the factors of $\\rho$ and $r$, the one-dimensional integral is a Hankel transform of order $n/2 - 1$.\n\n## Explanations\n\n### L1 human\n\nA radial function looks the same in every direction, like the ripples around a stone thrown into water.\nIts Fourier transform is radial too, so it is a function of one variable.\nThe formula computes it by a single integral over the radius, in which a Bessel function replaces the sine and cosine of the one-dimensional case.\n\n### L2 human\n\nThis reduction is the main tool for radial problems on $\\R^n$.\nThe transforms of the indicator of a ball, of the surface measure on a sphere and of the Bochner–Riesz multipliers are all computed this way.\nThe decay of Bessel functions, $J_\\nu(z) = O(z^{-1/2})$ as $z \\to \\infty$, has a direct consequence.\nThe Fourier transform of the surface measure on a sphere decays like $\\abs{\\xi}^{-(n-1)/2}$, a fact at the heart of restriction theory.\n",
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