  Functional times series have become an integral part of both functional data
and time series analysis. This paper deals with the functional autoregressive
model of order 1 and the autoregression bootstrap for smooth functions. The
regression operator is estimated in the framework developed by Ferraty and Vieu
[2004] and Ferraty et al. [2007] which is here extended to the double
functional case under an assumption of stationary ergodic data which dates back
to Laib and Louani [2010]. The main result of this article is the
characterization of the asymptotic consistency of the bootstrapped regression
operator.
