  This study develops an asymptotic theory for estimating the time-varying
characteristics of locally stationary functional time series (LSFTS). We
investigate a kernel-based method to estimate the time-varying covariance
operator and the time-varying mean function of an LSFTS. In particular, we
derive the convergence rate of the kernel estimator of the covariance operator
and associated eigenvalue and eigenfunctions and establish a central limit
theorem for the kernel-based locally weighted sample mean. As applications of
our results, we discuss methods for testing the equality of time-varying mean
functions in two functional samples.
