  Covariance function estimation is a fundamental task in multivariate
functional data analysis and arises in many applications. In this paper, we
consider estimating sparse covariance functions for high-dimensional functional
data, where the number of random functions p is comparable to, or even larger
than the sample size n. Aided by the Hilbert--Schmidt norm of functions, we
introduce a new class of functional thresholding operators that combine
functional versions of thresholding and shrinkage, and propose the adaptive
functional thresholding estimator by incorporating the variance effects of
individual entries of the sample covariance function into functional
thresholding. To handle the practical scenario where curves are partially
observed with errors, we also develop a nonparametric smoothing approach to
obtain the smoothed adaptive functional thresholding estimator and its binned
implementation to accelerate the computation. We investigate the theoretical
properties of our proposals when p grows exponentially with n under both fully
and partially observed functional scenarios. Finally, we demonstrate that the
proposed adaptive functional thresholding estimators significantly outperform
the competitors through extensive simulations and the functional connectivity
analysis of two neuroimaging datasets.
