  We study functional regression with random subgaussian design and real-valued
response. The focus is on the problems in which the regression function can be
well approximated by a functional linear model with the slope function being
"sparse" in the sense that it can be represented as a sum of a small number of
well separated "spikes". This can be viewed as an extension of now classical
sparse estimation problems to the case of infinite dictionaries. We study an
estimator of the regression function based on penalized empirical risk
minimization with quadratic loss and the complexity penalty defined in terms of
$L_1$-norm (a continuous version of LASSO). The main goal is to introduce
several important parameters characterizing sparsity in this class of problems
and to prove sharp oracle inequalities showing how the $L_2$-error of the
continuous LASSO estimator depends on the underlying sparsity of the problem.
