Continuous families of vectors that are orbits of a unitary group

We characterize functions u from the real line into a Hilbert space that are the orbits of a unitary group { U ( t ) } t ∈ R ; that is, u ( t ) = U ( t ) u ( 0 ) , for all real t. One of the characterizations is that u be the Fourier transform of a certain type of vector-valued measure Z; we then use our characterizations to construct Z from u.