|Abstract:|| If only fragments of functions are observable and none of these fragments cover the total domain, then it is impossible to estimate the covariance function over its total support. We propose a new functional PCA based prediction procedure that allows to reconstruct functional data from their fragmental observations under this challenging setup. By contrast to functional prediction models in the literature, we can proof the optimality of our prediction model without making use of the crucial, but unverifiable,orthogonality assumption on the prediction error. This also allows us to
identify situations under which our prediction procedure leads to a perfect reconstruction of the fragmental observations, i.e., without any prediction error. Finite sample properties are investigated through simulations and a
real data application.