  A bootstrap procedure for functional time series is proposed which exploits a
general vector autoregressive representation of the time series of Fourier
coefficients appearing in the Karhunen-Lo\`eve expansion of the functional
process. A double sieve-type bootstrap method is developed which avoids the
estimation of process operators and generates functional pseudo-time series
that appropriately mimic the dependence structure of the functional time series
at hand. The method uses a finite set of functional principal components to
capture the essential driving parts of the infinite dimensional process and a
finite order vector autoregressive process to imitate the temporal dependence
structure of the corresponding vector time series of Fourier coefficients. By
allowing the number of functional principal components as well as the
autoregressive order used to increase to infinity (at some appropriate rate) as
the sample size increases, a basic bootstrap central limit theorem is
established which shows validity of the bootstrap procedure proposed for
functional finite Fourier transforms. Some numerical examples illustrate the
good finite sample performance of the new bootstrap method proposed.
