  Functional data analyses typically proceed by smoothing, followed by
functional PCA. This paradigm implicitly assumes that rough variation is due to
nuisance noise. Nevertheless, relevant functional features such as
time-localised or short scale fluctuations may indeed be rough relative to the
global scale, but still smooth at shorter scales. These may be confounded with
the global smooth components of variation by the smoothing and PCA, potentially
distorting the parsimony and interpretability of the analysis. The goal of this
paper is to investigate how both smooth and rough variations can be recovered
on the basis of discretely observed functional data. Assuming that a functional
datum arises as the sum of two uncorrelated components, one smooth and one
rough, we develop identifiability conditions for the recovery of the two
corresponding covariance operators. The key insight is that they should possess
complementary forms of parsimony: one smooth and finite rank (large scale), and
the other banded and potentially infinite rank (small scale). Our conditions
elucidate the precise interplay between rank, bandwidth, and grid resolution.
Under these conditions, we show that the recovery problem is equivalent to
rank-constrained matrix completion, and exploit this to construct estimators of
the two covariances, without assuming knowledge of the true bandwidth or rank;
we study their asymptotic behaviour, and then use them to recover the smooth
and rough components of each functional datum by best linear prediction. As a
result, we effectively produce separate functional PCAs for smooth and rough
variation.
