  Inference via simultaneous confidence band is studied for stationary
covariance function of dense functional data. A two-stage estimation procedure
is proposed based on spline approximation, the first stage involving estimation
of all the individual trajectories and the second stage involving estimation of
the covariance function through smoothing the empirical covariance function.
The proposed covariance estimator is smooth and as efficient as the oracle
estimator when all individual trajectories are known. An asymptotic
simultaneous confidence band (SCB) is developed for the true covariance
function, and the coverage probabilities are shown to be asymptotically
correct. Simulation experiments are conducted on the numerical performance of
the proposed estimator and SCB. The proposed method is also illustrated by two
real data examples.
