  When dealing with very large datasets of functional data, survey sampling
approaches are useful in order to obtain estimators of simple functional
quantities, without being obliged to store all the data. We propose here a
Horvitz--Thompson estimator of the mean trajectory. In the context of a
superpopulation framework, we prove under mild regularity conditions that we
obtain uniformly consistent estimators of the mean function and of its variance
function. With additional assumptions on the sampling design we state a
functional Central Limit Theorem and deduce asymptotic confidence bands.
Stratified sampling is studied in detail, and we also obtain a functional
version of the usual optimal allocation rule considering a mean variance
criterion. These techniques are illustrated by means of a test population of
N=18902 electricity meters for which we have individual electricity consumption
measures every 30 minutes over one week. We show that stratification can
substantially improve both the accuracy of the estimators and reduce the width
of the global confidence bands compared to simple random sampling without
replacement.
