Abstract
Our aim is to estimate the unknown slope function in the functional linear model when the response $Y$ is real and the random function X is a second order stationary and periodic process. We obtain our estimator by minimizing a standard (and very simple) mean-square contrast on linear finite dimensional spaces spanned by trigonometric bases. The novelty of our approach is to provide a penalization procedure which allows to automatically select the adequate dimension, in a non-asymptotic point of view. In fact, we can show that our penalized estimator reaches the optimal (minimax) rate of convergence in the sense of the prediction error. We complete the theoretical results by a simulation study and a real data example which illustrate how the procedure works in practice.