Faster convergence rate for functional linear regression in reproducing kernel Hilbert spaces

Functional linear regression is in the centre of research attention involving curves as units of observations. We focus on functional linear regression in the framework of reproducing kernel Hilbert spaces studied in Cai and Yuan [Minimax and adaptive prediction for functional linear regression. J Am Stat Assoc. 2012;107(499):1201-1216]. We extend their theoretical result establishing faster convergence rate under stronger conditions which is reduced to existing results when the stronger condition is removed. In particular, our result corroborates the expectation that with smoother functions the convergence rate of the estimator is faster.