  Previous analysis of regularized functional linear regression in a
reproducing kernel Hilbert space (RKHS) typically requires the target function
to be contained in this kernel space. This paper studies the convergence
performance of divide-and-conquer estimators in the scenario that the target
function does not necessarily reside in the underlying RKHS. As a
decomposition-based scalable approach, the divide-and-conquer estimators of
functional linear regression can substantially reduce the algorithmic
complexities in time and memory. We develop an integral operator approach to
establish sharp finite sample upper bounds for prediction with
divide-and-conquer estimators under various regularity conditions of
explanatory variables and target function. We also prove the asymptotic
optimality of the derived rates by building the mini-max lower bounds. Finally,
we consider the convergence of noiseless estimators and show that the rates can
be arbitrarily fast under mild conditions.
