John's equation

John's equation is an ultrahyperbolic partial differential equation satisfied by the X-ray transform of a function. It is named after Fritz John.

Given a function ${\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} }$ with compact support the X-ray transform is the integral over all lines in ${\displaystyle \mathbb {R} ^{n}}$. We will parameterise the lines by pairs of points ${\displaystyle x,y\in \mathbb {R} ^{n}}$, ${\displaystyle x\neq y}$ on each line and define ${\displaystyle u}$ as the ray transform where

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