  This paper studies linear reconstruction of partially observed functional
data which are recorded on a discrete grid. We propose a novel estimation
approach based on approximate factor models with increasing rank taking into
account potential covariate information. Whereas alternative reconstruction
procedures commonly involve some preliminary smoothing, our method separates
the signal from noise and reconstructs missing fragments at once. We establish
uniform convergence rates of our estimator and introduce a new method for
constructing simultaneous prediction bands for the missing trajectories. A
simulation study examines the performance of the proposed methods in finite
samples. Finally, a real data application of temperature curves demonstrates
that our theory provides a simple and effective method to recover missing
fragments.
