  We study in this paper a smoothness regularization method for functional
linear regression and provide a unified treatment for both the prediction and
estimation problems. By developing a tool on simultaneous diagonalization of
two positive definite kernels, we obtain shaper results on the minimax rates of
convergence and show that smoothness regularized estimators achieve the optimal
rates of convergence for both prediction and estimation under conditions weaker
than those for the functional principal components based methods developed in
the literature. Despite the generality of the method of regularization, we show
that the procedure is easily implementable. Numerical results are obtained to
illustrate the merits of the method and to demonstrate the theoretical
developments.
