  The equations governing the flow of a viscous incompressible fluid around a
rigid body that performs a prescribed time-periodic motion with constant axes
of translation and rotation are investigated. Under the assumption that the
period and the angular velocity of the prescribed rigid-body motion are
compatible, and that the mean translational velocity is non-zero, existence of
a time-periodic solution is established. The proof is based on an appropriate
linearization, which is examined within a setting of absolutely convergent
Fourier series. Since the corresponding resolvent problem is ill-posed in
classical Sobolev spaces, a linear theory is developed in a framework of
homogeneous Sobolev spaces.
