  Functional linear discriminant analysis offers a simple yet efficient method
for classification, with the possibility of achieving a perfect classification.
Several methods are proposed in the literature that mostly address the
dimensionality of the problem. On the other hand, there is a growing interest
in interpretability of the analysis, which favors a simple and sparse solution.
In this work, we propose a new approach that incorporates a type of sparsity
that identifies non-zero sub-domains in the functional setting, offering a
solution that is easier to interpret without compromising performance. With the
need to embed additional constraints in the solution, we reformulate the
functional linear discriminant analysis as a regularization problem with an
appropriate penalty. Inspired by the success of $\ell_1$-type regularization at
inducing zero coefficients for scalar variables, we develop a new
regularization method for functional linear discriminant analysis that
incorporates an $L^1$-type penalty, $\int |f|$, to induce zero regions. We
demonstrate that our formulation has a well defined solution that contains zero
regions, achieving a functional sparsity in the sense of domain selection. In
addition, the misclassification probability of the regularized solution is
shown to converge to the Bayes error if the data are Gaussian. Our method does
not presume that the underlying function has zero regions in the domain, but
produces a sparse estimator that consistently estimates the true function
whether or not the latter is sparse. Numerical comparisons with existing
methods demonstrate this property in finite samples with both simulated and
real data examples.
