  It is more and more frequently the case in applications that the data we
observe come from one or more random variables taking values in an infinite
dimensional space, e.g. curves. The need to have tools adapted to the nature of
these data explains the growing interest in the field of functional data
analysis. The model we study in this paper assumes a linear dependence between
a quantity of interest and several covariates, at least one of which has an
infinite dimension. To select the relevant covariates in this context, we
investigate adaptations of the Lasso method. Two estimation methods are
defined. The first one consists in the minimization of a Group-Lasso criterion
on the multivariate functional space H. The second one minimizes the same
criterion but on a finite dimensional subspaces of H whose dimension is chosen
by a penalized least squares method. We prove oracle inequalities of sparsity
in the case where the design is fixed or random. To compute the solutions of
both criteria in practice, we propose a coordinate descent algorithm. A
numerical study on simulated and real data illustrates the behavior of the
estimators.
