This paper investigates optimal linear prediction for a random function in an infinite-dimensional Hilbert space. We analyze the mean square prediction error (MSPE) associated with a linear predictor, revealing that non-unique solutions that minimize the MSPE generally exist, and consistent estimation is often impossible even if a unique solution exists. However, this paper shows that it is still feasible to construct an asymptotically optimal linear operator, for which the empirical MSPE approaches the minimal achievable level. Remarkably, standard post-dimension reduction estimators, widely employed in the literature, serve as such estimators under minimal conditions. This finding affirms the use of standard post-dimension reduction estimators as a way to achieve the minimum MSPE without requiring a careful examination of various technical conditions commonly required in functional linear models.

翻译：本文研究无限维希尔伯特空间中随机函数的最优线性预测问题。我们分析了与线性预测器相关的均方预测误差（MSPE），揭示了通常存在使MSPE最小化的非唯一解，且即使存在唯一解，一致估计也往往难以实现。然而，本文证明构建渐近最优线性算子仍是可行的，使得经验MSPE能够逼近理论可达到的最小水平。值得注意的是，文献中广泛使用的标准降维后估计量在极弱条件下即可充当此类算子。这一发现确认了标准降维后估计量可作为实现最小MSPE的有效途径，无需仔细检验函数线性模型中常见的各类技术条件。