  We build up an asymptotic observability identity for the heat equation in the
whole space.
  It says that one can approximately recover a solution, through observing it
over some countable lattice points in the space and at one time.
  This asymptotic identity is a natural extension of the well-known
Shannon-Whittaker sampling theorem \cite{Shannon,Whittaker}.
  According to it, we obtain a kind of feedback null approximate
controllability for impulsively controlled heat equations.
  We also obtain a weak asymptotic observability identity with finitely many
observation lattice points. This identity holds only for some solutions to the
heat equation.
