It is named after German-American mathematician Fritz John.
with compact support the X-ray transform is the integral over all lines in
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
, 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.