Estimation of the Functional Linear Regression with Smoothing Splines

We consider functional linear regression where a real variable Y depends on a functional variable X. The functional coefficient of the model is estimated by means of smoothing splines. We derive the rates of convergence with respect to the semi-norm induced by the covariance operator of X, which comes to evaluate the error of prediction. These rates, which essentially depend on the smoothness of the function parameter and on the structure of the predictor, are shown to be optimal over a large class of functions parameters and distributions of the predictor.