  When the study variable is functional and storage capacities are limited or
transmission costs are high, selecting with survey sampling techniques a small
fraction of the observations is an interesting alternative to signal
compression techniques, particularly when the goal is the estimation of simple
quantities such as means or totals. We extend, in this functional framework,
model-assisted estimators with linear regression models that can take account
of auxiliary variables whose totals over the population are known. We first
show, under weak hypotheses on the sampling design and the regularity of the
trajectories, that the estimator of the mean function as well as its variance
estimator are uniformly consistent. Then, under additional assumptions, we
prove a functional central limit theorem and we assess rigorously a fast
technique based on simulations of Gaussian processes which is employed to build
asymptotic confidence bands. The accuracy of the variance function estimator is
evaluated on a real dataset of sampled electricity consumption curves measured
every half an hour over a period of one week.
