  Nonparametric estimation of the mean and covariance functions is ubiquitous
in functional data analysis and local linear smoothing techniques are most
frequently used. Zhang and Wang (2016) explored different types of asymptotic
properties of the estimation, which reveal interesting phase transition
phenomena based on the relative order of the average sampling frequency per
subject $T$ to the number of subjects $n$, partitioning the data into three
categories: ``sparse'', ``semi-dense'' and ``ultra-dense''. In an increasingly
available high-dimensional scenario, where the number of functional variables
$p$ is large in relation to $n$, we revisit this open problem from a
non-asymptotic perspective by deriving comprehensive concentration inequalities
for the local linear smoothers. Besides being of interest by themselves, our
non-asymptotic results lead to elementwise maximum rates of $L_2$ convergence
and uniform convergence serving as a fundamentally important tool for further
convergence analysis when $p$ grows exponentially with $n$ and possibly $T$.
With the presence of extra $\log p$ terms to account for the high-dimensional
effect, we then investigate the scaled phase transitions and the corresponding
elementwise maximum rates from sparse to semi-dense to ultra-dense functional
data in high dimensions. Finally, numerical studies are carried out to confirm
our established theoretical properties.
