We propose a new reconstruction operator that aims to recover the missing parts of a function given the observed parts. This new operator belongs to a new, very large class of functional operators which includes the classical regression operators as a special case. We show the optimality of our reconstruction operator and demonstrate that the usually considered regression operators generally cannot be optimal reconstruction operators. Our estimation theory allows for autocorrelated functional data and considers the practically relevant situation in which each of the $n$ functions is observed at $m$ discretization points. We derive rates of consistency for our nonparametric estimation procedures using a double asymptotic ($n\to\infty, m\to\infty$). For data situations, as in our real data application where $m$ is considerably smaller than $n$, we show that our functional principal components based estimator can provide better rates of convergence than any conventional nonparametric smoothing method.

中文翻译：

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关于部分观测功能数据的优化重建

我们提出了一种新的重建算子，旨在根据观察到的部分恢复函数的缺失部分。这个新算子属于一个新的非常大的函数算子类别，其中包括作为特例的经典回归算子。我们展示了我们的重构算子的最优性，并证明了通常考虑的回归算子通常不能是最优重构算子。我们的估计理论允许自相关函数数据，并考虑在 $m$ 离散点处观察到 $n$ 函数中的每一个的实际相关情况。我们使用双渐近 ($n\to\infty, m\to\infty$) 推导出非参数估计程序的一致性率。对于数据情况，如在我们的真实数据应用程序中，$m$ 远小于 $n$，