  In this paper, we propose a new regularization technique called "functional
SCAD". We then combine this technique with the smoothing spline method to
develop a smooth and locally sparse (i.e., zero on some sub-regions) estimator
for the coefficient function in functional linear regression. The functional
SCAD has a nice shrinkage property that enables our estimating procedure to
identify the null subregions of the coefficient function without over shrinking
the non-zero values of the coefficient function. Additionally, the smoothness
of our estimated coefficient function is regularized by a roughness penalty
rather than by controlling the number of knots. Our method is more
theoretically sound and is computationally simpler than the other available
methods. An asymptotic analysis shows that our estimator is consistent and can
identify the null region with the probability tending to one. Furthermore,
simulation studies show that our estimator has superior numerical performance.
Finally, the practical merit of our method is demonstrated on two real
applications.
