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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $S^{n-1} = \\set{\\theta \\in \\R^n : \\abs{\\theta} = 1}$ be the unit [[sphere|P9CM4GQ2T0]], let $\\sigma$ be its surface measure, and let $\\omega_{n-1} = \\sigma(S^{n-1})$.\nLet $f \\colon \\R^n \\to \\C$ be continuous.\nThe *spherical mean transform* of $f$ is the function on $\\R^n \\times [0, \\infty)$ given by\n$$(M f)(x, r) = \\frac{1}{\\omega_{n-1}} \\int_{S^{n-1}} f(x + r\\theta)\\, d\\sigma(\\theta).$$\n\n## Remarks\n\n$M f(x, r)$ is the average of $f$ over the sphere of center $x$ and radius $r$, and $M f(x, 0) = f(x)$.\nIf $f$ is of class $C^2$, then $M f$ satisfies the Darboux equation\n$$\\Delta_x M f(x, r) = \\left( \\frac{\\partial^2}{\\partial r^2} + \\frac{n - 1}{r} \\frac{\\partial}{\\partial r} \\right) M f(x, r), \\qquad r \u003e 0.$$\n\nExample: $f(x) = \\abs{x}^2$.\nSince $\\abs{x + r\\theta}^2 = \\abs{x}^2 + 2r\\, x \\cdot \\theta + r^2$ and $\\theta \\mapsto x \\cdot \\theta$ has mean $0$ over the sphere, $M f(x, r) = \\abs{x}^2 + r^2$.\nBoth sides of the Darboux equation equal $2n$.\n\nNon-example: $f(y) = \\abs{y}^{1-n}$ for $y \\ne 0$ and $f(0) = 0$ is not continuous.\nIts mean over the sphere of center $x \\ne 0$ and radius $r = \\abs{x}$, which passes through $0$, is $+\\infty$.\n\nThe function $f$ may also be only locally integrable, and then $M f(x, r)$ is defined for almost every $(x, r)$.\n",
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