  In a recent work, Unruh showed that Hawking radiation is unaffected by a
truncation of free field theory at the Planck scale. His analysis was performed
numerically and based on a hydrodynamical model. In this work, by analytical
methods, the mathematical and physical origin of Unruh's result is revealed. An
alternative truncation scheme which may be more appropriate for black hole
physics is proposed and analyzed. In both schemes the thermal Hawking radiation
remains unaffected even though transplanckian energies no longer appear. The
universality of this result is explained by working in momentum space. In that
representation, in the presence of a horizon, the d'Alembertian equation
becomes a singular first order equation. In addition, the boundary conditions
corresponding to vacuum before the black hole formed are that the in--modes
contain positive momenta only. Both properties remain valid when the spectrum
is truncated and they suffice to obtain Hawking radiation.
