  In this paper, we propose methods for functional predictor selection and the
estimation of smooth functional coefficients simultaneously in a
scalar-on-function regression problem under high-dimensional multivariate
functional data setting. In particular, we develop two methods for functional
group-sparse regression under a generic Hilbert space of infinite dimension. We
show the convergence of algorithms and the consistency of the estimation and
the selection (oracle property) under infinite-dimensional Hilbert spaces.
Simulation studies show the effectiveness of the methods in both the selection
and the estimation of functional coefficients. The applications to the
functional magnetic resonance imaging (fMRI) reveal the regions of the human
brain related to ADHD and IQ.
