Reproduction Capabilities of Penalized Hyperbolic-polynomial Splines

02/14/2022

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by   Rosanna Campagna, et al.

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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 e^α x, e^-α x and linear functions multiplied by these exponentials, xe^+α x and xe^-α x. Here, we show that these type of penalized splines reproduce function in the space {e^-α x, x e^-α x}, that is they fit exponential data exactly. Moreover, we show that they conserve the first and second 'exponential' moments.