some asymptotic properties of functional linear regression model withpoints of impact

The functional linear regression model with points of impact (poi) is arecent augmentation of the classical functional linear model with many practically importantapplications. it is assumed that there exists an unknown number of impactpoints that is, discrete observation times where the corresponding functional valuespossess significant influences on the response variable. in this paper, we obtain someasymptotic properties of the model that can be used for further statistical inferencesabout the response variable. specifically, the rates of convergence for eigenfunctionsestimates of the predictor covariance operator evaluated at the impact points estimatesare derived. these are important results, because we do not have true eigenfunctionsand impact points in applications and we have to use their estimates instead.