  We prove various uniqueness results from null infinity, for linear waves on
asymptotically flat space-times. Assuming vanishing of the solution to infinite
order on suitable parts of future and past null infinities, we derive that the
solution must vanish in an open set in the interior. We find that the parts of
infinity where we must impose a vanishing condition depend strongly on the
background geometry. In particular, for backgrounds with positive mass (such as
Schwarzschild or Kerr), the required assumptions are much weaker than the ones
in the Minkowski space-time. The results are nearly optimal in many respects.
They can be considered analogues of uniqueness from infinity results for second
order elliptic operators. This work is partly motivated by questions in general
relativity.
