  We consider the problem of predicting a real random variable from a
functional explanatory variable. The problem is attacked by mean of
nonparametric kernel approach which has been recently adapted to this
functional context. We derive theoretical results by giving a deep asymptotic
study of the behaviour of the estimate, including mean squared convergence
(with rates and precise evaluation of the constant terms) as well as asymptotic
distribution. Practical use of these results are relying on the ability to
estimate these constants. Some perspectives in this direction are discussed
including the presentation of a functional version of bootstrapping ideas.
