  This paper introduces a new operator relevant to input-output analysis of
flows in a statistically steady regime far from the steady base flow: the mean
resolvent $\mathbf{R}_0$. It is defined as the operator predicting, in the
frequency domain, the mean linear response to forcing of the time-varying base
flow. As such, it provides the statistically optimal linear time-invariant
approximation of the input-output dynamics, which may be useful, for instance,
in flow control applications. Theory is developed for the periodic case. The
poles of the operator are shown to correspond to the Floquet exponents of the
system, including purely imaginary poles at multiples of the fundamental
frequency. In general, evaluating mean transfer functions from data requires
averaging the response to many realizations of the same input. However, in the
specific case of harmonic forcings, we show that the mean transfer functions
may be identified without averaging: an observation referred to as `dynamic
linearity' in the literature (Dahan et al., 2012). For incompressible flows in
the weakly unsteady limit, i.e. when amplification of perturbations by the
unsteady part of the periodic Jacobian is small compared to amplification by
the mean Jacobian, the mean resolvent $\mathbf{R}_0$ is well-approximated by
the well-known resolvent operator about the mean-flow. Although the theory
presented in this paper only extends to quasiperiodic flows, the definition of
$\mathbf{R}_0$ remains meaningful for flows with continuous or mixed spectra,
including turbulent flows. Numerical evidence supports the close connection
between the two resolvent operators in quasiperiodic, chaotic and stochastic
two-dimensional incompressible flows.
