  We study regression models for the situation where both dependent and
independent variables are square-integrable stochastic processes. Questions
concerning the definition and existence of the corresponding functional linear
regression models and some basic properties are explored for this situation. We
derive a representation of the regression parameter function in terms of the
canonical components of the processes involved. This representation establishes
a connection between functional regression and functional canonical analysis
and suggests alternative approaches for the implementation of functional linear
regression analysis. A specific procedure for the estimation of the regression
parameter function using canonical expansions is proposed and compared with an
established functional principal component regression approach. As an example
of an application, we present an analysis of mortality data for cohorts of
medflies, obtained in experimental studies of aging and longevity.
