  A mathematical model for variable selection in functional regression models
with scalar response is proposed. By "variable selection" we mean a procedure
to replace the whole trajectories of the functional explanatory variables with
their values at a finite number of carefully selected instants (or "impact
points"). The basic idea of our approach is to use the Reproducing Kernel
Hilbert Space (RKHS) associated with the underlying process, instead of the
more usual L2[0,1] space, in the definition of the linear model. This turns out
to be especially suitable for variable selection purposes, since the
finite-dimensional linear model based on the selected "impact points" can be
seen as a particular case of the RKHS-based linear functional model. In this
framework, we address the consistent estimation of the optimal design of impact
points and we check, via simulations and real data examples, the performance of
the proposed method.
