  Function-on-function linear regression is important for understanding the
relationship between the response and the predictor that are both functions. In
this article, we propose a reproducing kernel Hilbert space approach to
function-on-function linear regressionvia the penalised least square,
regularized by the thin-plate spline smoothness penalty. The minimax optimal
convergence rate of our estimator of the coefficient function is studied. We
derive the Bahadur representation, which allows us to propose statistical
inference methods using bootstrap and the convergence of Banach-valued random
variables in the sup-norm. We illustrate our method and verify our theoretical
results via simulated data experiments and a real data example.
