On prediction error in functional linear regression

We consider a regression setting where the response is a scalar and the predictor is a random function. Many fields of applications are concerned with such data, for example chemometrics. When researchers are faced with the estimation of a functional (infinite dimensional) coefficient, they reduce the dimension by projecting the weight function onto a lower dimensional space. We derive an upper bound for the mean squared error of prediction when the choice of the lower dimensional space is guided by the smoothness of the regression function.