Convergence of eigenfunctions with diverging index is essential in nearly all methods based on functional principal components analysis. The main goal of this work is to establish the unified theory for such eigencomponents in different types of convergence based on discretely observed functional data. We obtain the moment bounds for eigenfunctions and eigenvalues for a wide range of the sampling rate and show that under some mild assumptions, the $\mathcal{L}^{2}$ bound of eigenfunctions estimator with diverging indices is optimal in the minimax sense as if the curves are fully observed. This is the first attempt at obtaining an optimal rate for eigenfunctions with diverging index for discretely observed functional data. We propose a double truncation technique in handling the uniform convergence of function data and establish the uniform convergence of covariance function as well as the eigenfunctions for all sampling scheme under mild assumptions. The technique route proposed in this work provides a new tool in handling the perturbation series with discretely observed functional data and can be applied in most problems based on functional principal components analysis and models involving inverse issue.

翻译：在基于功能性主要组成部分分析的几乎所有方法中,对基于功能性主要组成部分分析的几乎所有方法都至关重要。这项工作的主要目标是,根据离散观测功能数据,为不同类型趋同的不同类型类同中的此类乙源成分确立统一理论;我们为广泛范围的取样率获得关于乙源功能和乙源值的时标,并表明,根据一些轻度假设,在有差异性主要组成部分的测算器中,美元=mathcal{L ⁇ 2}美元是含有不同指数的测算器的最佳方法,与完全观察到曲线的情况一样。这是首次尝试为离散观测功能性数据获得具有不同指数的这种类同源成分的最佳比率。我们提出了一种处理功能性数据统一趋同的双重调试算法,并在一些轻度假设下确定所有采样方法的共变系数和均匀值的统一。这项工作提出的技术途径提供了一种新工具,用以用独立观测到的功能性数据处理扰动序列,可以应用于大多数涉及功能性分析问题的模型。