  Within the framework of functional data analysis, we develop principal
component analysis for periodically correlated time series of functions. We
define the components of the above analysis including periodic, operator-valued
filters, score processes and the inversion formulas. We show that these objects
are defined via convergent series under a simple condition requiring
summability of the Hilbert-Schmidt norms of the filter coefficients, and that
they poses optimality properties.
  We explain how the Hilbert space theory reduces to an approximate
finite-dimensional setting which is implemented in a custom build R package. A
data example and a simulation study show that the new methodology is superior
to existing tools if the functional time series exhibit periodic
characteristics.
