some asymptotic properties of functional linear regression model with points of impact

The functional linear regression model with points of impact is a recent augmentation of the classical functional linear model with many practically important applications‎. ‎it is assumed that there exists an unknown number of impact points‎, ‎that is discrete observation times where the corresponding functional values possess significant influences on the response variable‎. ‎in this paper‎, ‎we obtain some asymptotic properties of the model that can be used for further statistical inferences about the response variable‎. ‎specifically‎, ‎rates of convergence for eigenfunctions estimates of the predictor covariance operator evaluated at the impact points estimates are derived‎. ‎these are important results‎, ‎because we do not have true eigenfunctions and impact points in applications and we have to use their estimates instead.