  Functional data analysis is an important research field in statistics which
treats data as random functions drawn from some infinite-dimensional functional
space, and functional principal component analysis (FPCA) based on
eigen-decomposition plays a central role for data reduction and representation.
After nearly three decades of research, there remains a key problem unsolved,
namely, the perturbation analysis of covariance operator for diverging number
of eigencomponents obtained from noisy and discretely observed data. This is
fundamental for studying models and methods based on FPCA, while there has not
been substantial progress since Hall, M\"uller and Wang (2006)'s result for a
fixed number of eigenfunction estimates. In this work, we aim to establish a
unified theory for this problem, obtaining upper bounds for eigenfunctions with
diverging indices in both the $\mathcal{L}^2$ and supremum norms, and deriving
the asymptotic distributions of eigenvalues for a wide range of sampling
schemes. Our results provide insight into the phenomenon when the
$\mathcal{L}^{2}$ bound of eigenfunction estimates with diverging indices is
minimax optimal as if the curves are fully observed, and reveal the transition
of convergence rates from nonparametric to parametric regimes in connection to
sparse or dense sampling. We also develop a double truncation technique to
handle the uniform convergence of estimated covariance and eigenfunctions. The
technical arguments in this work are useful for handling the perturbation
series with noisy and discretely observed functional data and can be applied in
models or those involving inverse problems based on FPCA as regularization,
such as functional linear regression.
