  As with classic statistics, functional regression models are invaluable in
the analysis of functional data. While there are now extensive tools with
accompanying theory available for linear models, there is still a great deal of
work to be done concerning nonlinear models for functional data. In this work
we consider the Additive Function-on-Function Regression model, a type of
nonlinear model that uses an additive relationship between the functional
outcome and functional covariate. We present an estimation methodology built
upon Reproducing Kernel Hilbert Spaces, and establish optimal rates of
convergence for our estimates in terms of prediction error. We also discuss
computational challenges that arise with such complex models, developing a
representer theorem for our estimate as well as a more practical and
computationally efficient approximation. Simulations and an application to
Cumulative Intraday Returns around the 2008 financial crisis are also provided.
