  In functional linear regression, the slope ``parameter'' is a function.
Therefore, in a nonparametric context, it is determined by an infinite number
of unknowns. Its estimation involves solving an ill-posed problem and has
points of contact with a range of methodologies, including statistical
smoothing and deconvolution. The standard approach to estimating the slope
function is based explicitly on functional principal components analysis and,
consequently, on spectral decomposition in terms of eigenvalues and
eigenfunctions. We discuss this approach in detail and show that in certain
circumstances, optimal convergence rates are achieved by the PCA technique. An
alternative approach based on quadratic regularisation is suggested and shown
to have advantages from some points of view.
