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 : R n → R with compact support the X-ray transform is the integral over all lines in R n . We will parameterise the lines by pairs of points x , y ∈ R n , x ≠ y on each line and define u as the ray transform where

Such functions u are characterized by John's equations

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

where n ≥ 2 , such that the quadratic form

can be reduced by a linear change of variables to the form

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.