  We show that the fractional Laplacian can be viewed as a Dirichlet-to-Neumann
map for a degenerate hyperbolic problem, namely, the wave equation with an
additional diffusion term that blows up at time zero. A solution to this wave
extension problem is obtained from the Schr\"odinger group by means of an
oscillatory subordination formula, which also allows us to find kernel
representations for such solutions. Asymptotics of related oscillatory
integrals are analysed in order to determine the correct domains for initial
data in the general extension problem involving non-negative self-adjoint
operators. An alternative approach using Bessel functions is also described.
