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

$u(x,y)=\int \limits _{-\infty }^{\infty }f(x+t(y-x))dt.$ Such functions $u$ are characterized by John's equations

${\frac {\partial ^{2}u}{\partial x_{i}\partial y_{j}}}-{\frac {\partial ^{2}u}{\partial y_{i}\partial x_{j}}}=0$ which is proved by Fritz John for dimension three and by Kurusa for higher dimensions.

In three-dimensional x-ray computerized tomography John's equation can be solved to fill in missing data, for example where the data is obtained from a point source traversing a curve, typically a helix.

More generally an ultrahyperbolic partial differential equation (a term coined by Richard Courant) is a second order partial differential equation of the form

$\sum \limits _{i,j=1}^{2n}a_{ij}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}+\sum \limits _{i=1}^{2n}b_{i}{\frac {\partial u}{\partial x_{i}}}+cu=0$ where $n\geq 2$ , such that the quadratic form

$\sum \limits _{i,j=1}^{2n}a_{ij}\xi _{i}\xi _{j}$ can be reduced by a linear change of variables to the form

$\sum \limits _{i=1}^{n}\xi _{i}^{2}-\sum \limits _{i=n+1}^{2n}\xi _{i}^{2}.$ It is not possible to arbitrarily specify the value of the solution on a non-characteristic hypersurface. John's paper however does give examples of manifolds on which an arbitrary specification of u can be extended to a solution.