  This work introduces a variant of resolvent analysis that identifies forcing
and response modes that are sparse in both space and time. This is achieved
through the use of a sparse principal component analysis (PCA) algorithm, which
formulates the associated optimization problem as a nonlinear eigenproblem that
can be solved with an inverse power method. We apply this method to parallel
shear flows, both in the case where we assume Fourier modes in time (as in
standard resolvent analysis) and obtain spatial localization, and where we
allow for temporally-sparse modes through the use of a linearized Navier-Stokes
operator discretized in both space and time. Appropriate choice of desired mode
sparsity allows for the identification of structures corresponding to high
amplification that are localized in both space and time. We report on the
similarities and differences between these structures and those from standard
methods of analysis. After validating this space-time resolvent analysis on
statistically-stationary channel flow, we next implement the methodology on a
time-periodic Stokes boundary layer, demonstrating the applicability of the
approach to non-statistically-stationary systems.
