  We consider the problem of estimating the slope parameter in functional
linear regression, where scalar responses Y1,...,Yn are modeled in dependence
of second order stationary random functions X1,...,Xn. An orthogonal series
estimator of the functional slope parameter with additional thresholding in the
Fourier domain is proposed and its performance is measured with respect to a
wide range of weighted risks covering as examples the mean squared prediction
error and the mean integrated squared error for derivative estimation. In this
paper the minimax optimal rate of convergence of the estimator is derived over
a large class of different regularity spaces for the slope parameter and of
different link conditions for the covariance operator. These general results
are illustrated by the particular example of the well-known Sobolev space of
periodic functions as regularity space for the slope parameter and the case of
finitely or infinitely smoothing covariance operator.
