  Many scientific studies collect data where the response and predictor
variables are both functions of time, location, or some other covariate.
Understanding the relationship between these functional variables is a common
goal in these studies. Motivated from two real-life examples, we present in
this paper a function-on-function regression model that can be used to analyze
such kind of functional data. Our estimator of the 2D coefficient function is
the optimizer of a form of penalized least squares where the penalty enforces a
certain level of smoothness on the estimator. Our first result is the
Representer Theorem which states that the exact optimizer of the penalized
least squares actually resides in a data-adaptive finite dimensional subspace
although the optimization problem is defined on a function space of infinite
dimensions. This theorem then allows us an easy incorporation of the Gaussian
quadrature into the optimization of the penalized least squares, which can be
carried out through standard numerical procedures. We also show that our
estimator achieves the minimax convergence rate in mean prediction under the
framework of function-on-function regression. Extensive simulation studies
demonstrate the numerical advantages of our method over the existing ones,
where a sparse functional data extension is also introduced. The proposed
method is then applied to our motivating examples of the benchmark Canadian
weather data and a histone regulation study.
