Optimal estimation of functional linear models

Observations which are realizations from some continuous process are frequent in sciences, engineering, economics, and other fields. We consider a linear model where the responses are random functions in a suitable Sobolev space. The process can't be observed directly. With smoothing procedures from the original data, both the response curves and their derivatives can be reconstructed, even separately. From these functions we estimate the vector of functional parameters and we prove a functional version of the Gauss-Markov theorem. We also obtain experimental designs which are optimal for the estimation of the considered model. The advantages of this theory are finally showed in a real data-set.