  We extend results of Caffarelli--Silvestre and Stinga--Torrea regarding a
characterization of fractional powers of differential operators via an
extension problem. Our results apply to generators of integrated families of
operators, in particular to infinitesimal generators of bounded $C_0$
semigroups and operators with purely imaginary symbol. We give integral
representations to the extension problem in terms of solutions to the heat
equation and the wave equation.
