The advent of P-splines, first introduced by Eilers and Marx in 2010 (see \cite{4}), has led to important developments in data regression through splines. With the aim of generalizing polynomial P-splines, in \cite{1} we have recently defined a model of penalized regression spline, called HP-spline, in which polynomial B-splines are replaced by hyperbolic-polynomial bell-shaped basis functions, and a suitably tailored penalization term replaces the classical second-order forward difference operator. HP-splines inherit from P-splines all model advantages and extend some of them. Indeed, they separate the data from the spline knots -so avoiding overfitting and boundary effects-, exactly fit exponential data, and conserve two type of \lq exponential\rq \, moments.
HP-splines are particularly interesting in applications that require analysis and forecasting of data with exponential trends: the starting idea of this work is the definition of a polynomial-exponential smoothing spline model to be used in the framework of the Laplace transform inversion as done in \cite{2,3}. The talk discusses the existence, uniqueness, and reproduction properties of HP-splines, and provides several examples supporting their effective usage in data analysis.

 \paragraph{Joint work with:} Costanza Conti.
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\bibitem{1} C.~Conti, R.~Campagna.\newblock Penalized exponential-polynomial splines. \newblock {\em Appl. Math. Letters}, 118, (2021) 107--159
\bibitem{2} R.~Campagna, C.~Conti, S.~Cuomo.\newblock Computational Error Bounds for Laplace Transform Inversion Based on Smoothing Splines. \newblock {\em Appl. Math. Comput.}, 383, (2020) 125--376 \bibitem{3} R.~Campagna, C.~Conti, S.~Cuomo.\newblock Smoothing exponential-polynomial splines for multiexponential decay data. \newblock {\em Dolomites Research note on Approximation}, (2019) 86--10. \bibitem{4} P.H.C.~Eilers and B.D.~Marx.\newblock Splines, knots, and penalties. \newblock {\em WIREs Comp. Stat.}, 2, (2010) 637-653.

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