The functional linear model is a regression model in which the explanatory variable is a continuous time process observed in a closed interval of R: Hence, the "vector of parameters" to be estimated belongs to the infinite dimensional space of R-valued operators defined on a space of functions. We propose here two estimators of the functional parameter of such a model by means of spline functions. These estimators take into account the dimensionality problem and we prove their consistency. The first one relies on a truncated functional principal components analysis and the second is based on penalized regression splines. These estimators are compared by means of simulations and applied to explain winter wheat yield with respect to climatic variations.

We analyze in a regression setting the link between a scalar response and a functional predictor by means of a Functional Generalized Linear Model. We first give a theoretical framework and then discuss identifiability of the model. The functional coefficient of the model is estimated via penalized likelihood with spline approximation. The L² rate of convergence of this estimator is given under smoothness assumption on the functional coefficient. Heuristic arguments show how these rates may be improved for some particular frame-works.

Electrophoresis is a biochemical process widely used in life sciences and genetics. Recovering information on weights and proportions of molecules from electrophoresis signals may be viewed as a linear inverse problem in the context of a regression setup. The target function to be estimated is known to be positive so that a positive estimator based on discretized Dirac functions is proposed. The methodology is to use boosting method for the target function with respect to discretized Dirac functions. The method is illustrated on simulated data and applied to electrophoresis.