Reproduction capabilities of penalized hyperbolic-polynomial splines

This paper investigates two important analytical properties of hyperbolic-polynomial penalized splines, HP-splines for short. HP-splines, obtained by combining a special type difference penalty with B-splines (HB-splines), were recently introduced the authors as generalization P-splines. HB-splines are bell-shaped basis functions consisting segments made real exponentials eαx,e−αx and linear multiplied these exponentials, xe+αx xe−αx. Here, we show that types splines reproduce in space {e−αx,xe−αx}, is they fit exponential data exactly. Moreover, conserve first second ‘exponential’ moments.