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 of difference penalty with hyperbolic-polynomial B-splines (HB-splines), were recently introduced by the authors as a generalization of P-splines. HB-splines are bell-shaped basis functions consisting of segments made of real exponentials exp(αx), exp(−αx), and linear functions multiplied by these exponentials, x exp(αx) and x exp(−αx). Here, we show that these types of penalized splines reproduce functions in the space {exp(-αx), x exp(−αx)}, that is they fit exponential data exactly. Moreover, we show that they conserve the first and second ‘exponential’ moments.