Optimal Rates for Functional Linear Regression with General Regularization

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\"older type source condition, which generalizes and sharpens all the known results in the literature.