Functional linear regression is one of the fundamental and well-studied methods in functional data analysis. In this work, we investigate the functional linear regression model within the context of reproducing kernel Hilbert space by employing general spectral regularization to approximate the slope function with certain smoothness assumptions. We establish optimal convergence rates for estimation and prediction errors associated with the proposed method under a H\"{o}lder type source condition, which generalizes and sharpens all the known results in the literature.

翻译：函数线性回归是函数数据分析中基础且得到充分研究的方法之一。在本工作中，我们通过采用一般谱正则化方法，在再生核希尔伯特空间的框架下研究函数线性回归模型，并在特定光滑性假设下逼近斜率函数。在Hölder型源条件假设下，我们为所提方法建立了估计误差与预测误差的最优收敛率，该结果推广并锐化了文献中所有已知结论。