  In functional data analysis (FDA), covariance function is fundamental not
only as a critical quantity for understanding elementary aspects of functional
data but also as an indispensable ingredient for many advanced FDA methods.
This paper develops a new class of nonparametric covariance function estimators
in terms of various spectral regularizations of an operator associated with a
reproducing kernel Hilbert space. Despite their nonparametric nature, the
covariance estimators are automatically positive semi-definite without any
additional modification steps. An unconventional representer theorem is
established to provide a finite dimensional representation for this class of
covariance estimators, which leads to a closed-form expression of the
corresponding $L^2$ eigen-decomposition. Trace-norm regularization is
particularly studied to further achieve a low-rank representation, another
desirable property which leads to dimension reduction and is often needed in
advanced FDA approaches. An efficient algorithm is developed based on the
accelerated proximal gradient method. This resulted estimator is shown to enjoy
an excellent rate of convergence under both fixed and random designs. The
outstanding practical performance of the trace-norm-regularized covariance
estimator is demonstrated by a simulation study and the analysis of a traffic
dataset.
