Abstract: Smoothness penalties are efficient regularization and dimension reduction tools for functional regressions. However, for spiky functional data observed on a dense grid, the coefficient function in a functional regression can be spiky and, hence, the smoothness regularization is inefficient and leads to over-smoothing. We propose a novel approach to fit the function-on-function regression model by viewing the spiky coefficient functions as derivatives of smooth auxiliary functions. Compared with the smoothness regularization or sparsity regularization imposed directly on the spiky coefficient function in existing methods, imposing smoothness regularization on the smooth auxiliary functions can more efficiently reduce the dimension and improve the performance of the fitted model. Using the estimated smooth auxiliary functions and taking derivatives, we can fit the model and make predictions. Simulation studies and real-data applications show that compared with existing methods, the new method can greatly improve model performance when the coefficient function is spiky, and performs similarly well when the coefficient function is smooth.

Key words and phrases: Auxiliary function, derivative, function-on-function regression, smoothness regularization, spiky functional data.