Nonparametric estimation of mean and covariance functions based on discretely observed data is important in functional data analysis. In this talk, we propose a framelet block-thresholding method for estimating mean and covariance functions from discretely sampled noisy observations. Estimated convergence rates are established for all types of sampling schemes. In particular, the results reveal a phase transition phenomenon related to the number of observations on each curve. The procedures are adaptive in automatically adjusting the smoothness properties of the underlying mean and covariance functions. In contrast, theoretical results for other smoothing methods hold in the setting where smoothness parameters are assumed to be known, since the regularization parameters of estimators that depend on smoothness properties need to be chosen carefully. Simulation studies and real data examples are provided to offer empirical support for the theoretical results. A comparison with other methods demonstrates that the proposed method outperforms in adaptivity.

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