  This work aims at performing Functional Principal Components Analysis (FPCA)
with Horvitz-Thompson estimators when the observations are curves collected
with survey sampling techniques. One important motivation for this study is
that FPCA is a dimension reduction tool which is the first step to develop
model assisted approaches that can take auxiliary information into account.
FPCA relies on the estimation of the eigenelements of the covariance operator
which can be seen as nonlinear functionals. Adapting to our functional context
the linearization technique based on the influence function developed by
Deville (1999), we prove that these estimators are asymptotically design
unbiased and consistent. Under mild assumptions, asymptotic variances are
derived for the FPCA' estimators and consistent estimators of them are
proposed. Our approach is illustrated with a simulation study and we check the
good properties of the proposed estimators of the eigenelements as well as
their variance estimators obtained with the linearization approach.
