  Suppose that $n$ statistical units are observed, each following the model
$Y(x_j)=m(x_j)+ \epsilon(x_j),\, j=1,...,N,$ where $m$ is a regression
function, $0 \leq x_1 <...<x_N \leq 1$ are observation times spaced according
to a sampling density $f$, and $\epsilon$ is a continuous-time error process
having mean zero and regular covariance function. Considering the local
polynomial estimation of $m$ and its derivatives, we derive asymptotic
expressions for the bias and variance as $n,N\to\infty$. Such results are
particularly relevant in the context of functional data where essential
information is contained in the derivatives. Based on these results, we deduce
optimal sampling densities, optimal bandwidths and asymptotic normality of the
estimator. Simulations are conducted in order to compare the performances of
local polynomial estimators based on exact optimal bandwidths, asymptotic
optimal bandwidths, and cross-validated bandwidths.
