  This paper develops a framework for the estimation of the functional mean and
the functional principal components when the functions form a random field.
More specifically, the data we study consist of curves
$X(\mathbf{s}_k;t),t\in[0,T]$, observed at spatial points
$\mathbf{s}_1,\mathbf{s}_2,\ldots,\mathbf{s}_N$. We establish conditions for
the sample average (in space) of the $X(\mathbf{s}_k)$ to be a consistent
estimator of the population mean function, and for the usual empirical
covariance operator to be a consistent estimator of the population covariance
operator. These conditions involve an interplay of the assumptions on an
appropriately defined dependence between the functions $X(\mathbf{s}_k)$ and
the assumptions on the spatial distribution of the points $\mathbf{s}_k$. The
rates of convergence may be the same as for i.i.d. functional samples, but
generally depend on the strength of dependence and appropriately quantified
distances between the points $\mathbf{s}_k$. We also formulate conditions for
the lack of consistency.
