  For the group O(p,q) we give a new construction of its minimal unitary
representation via Euclidean Fourier analysis.
  This is an extension of the q = 2 case, where the representation is the mass
zero, spin zero representation realized in a Hilbert space of solutions to the
wave equation. The group O(p,q) acts as the Moebius group of conformal
transformations on R^{p-1, q-1}, and preserves a space of solutions of the
ultrahyperbolic Laplace equation on R^{p-1, q-1}.
  We construct in an intrinsic and natural way a Hilbert space of
ultrahyperbolic solutions so that O(p,q) becomes a continuous irreducible
unitary representation in this Hilbert space.
  We also prove that this representation is unitarily equivalent to the
representation on L^2(C), where C is the conical subvariety of the nilradical
of a maximal parabolic subalgebra obtained by intersecting with the minimal
nilpotent orbit in the Lie algebra of O(p,q).
