  We present a new methodology for simultaneous variable selection and
parameter estimation in function-on-scalar regression with an ultra-high
dimensional predictor vector. We extend the LASSO to functional data in both
the $\textit{dense}$ functional setting and the $\textit{sparse}$ functional
setting. We provide theoretical guarantees which allow for an exponential
number of predictor variables. Simulations are carried out which illustrate the
methodology and compare the sparse/functional methods. Using the Framingham
Heart Study, we demonstrate how our tools can be used in genome-wide
association studies, finding a number of genetic mutations which affect blood
pressure and are therefore important for cardiovascular health.
