  Functional Principal Component Analysis is a reference method for dimension
reduction of curve data. Its theoretical properties are now well understood in
the simplified case where the sample curves are fully observed without noise.
However, functional data are noisy and necessarily observed on a finite
discretization grid. Common practice consists in smoothing the data and then to
compute the functional estimates, but the impact of this denoising step on the
procedure's statistical performance are rarely considered. Here we prove new
convergence rates for functional principal component estimators. We introduce a
double asymptotic framework: one corresponding to the sampling size and a
second to the size of the grid. We prove that estimates based on projection
onto histograms show optimal rates in a minimax sense. Theoretical results are
illustrated on simulated data and the method is applied to the visualization of
genomic data.
