Reproduction Capabilities of Penalized Hyperbolic-polynomial Splines

TL;DR

This paper analyzes the analytical properties of hyperbolic-polynomial penalized splines, demonstrating their ability to reproduce exponential functions and conserve exponential moments, which enhances understanding of their approximation capabilities.

Contribution

It introduces new theoretical insights into HP-splines, showing they can exactly reproduce specific exponential functions and preserve exponential moments.

Findings

HP-splines reproduce functions in e^{-\u03b1 x}, x e^{- x}

They conserve the first and second exponential moments

HP-splines generalize P-splines with enhanced analytical properties.

Abstract

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