  Accurate curve forecasting is of vital importance for policy planning,
decision making and resource allocation in many engineering and industrial
applications. In this paper we establish a theoretical foundation for the
optimal short-term linear prediction of non-stationary functional or curve time
series with smoothly time-varying data generating mechanisms. The core of this
work is to establish a unified functional auto-regressive approximation result
for a general class of locally stationary functional time series. A double
sieve expansion method is proposed and theoretically verified for the
asymptotic optimal forecasting. A telecommunication traffic data set is used to
illustrate the usefulness of the proposed theory and methodology.
