Polyhyperbolic Cardinal Splines

TL;DR

This paper introduces hyperbolic splines, solutions to a specific differential equation, and develops their interpolation properties, expanding the mathematical understanding of these functions.

Contribution

It provides a detailed analysis of hyperbolic splines, including their fundamental functions and key properties, which were not previously well-understood.

Findings

Derived the fundamental function of hyperbolic splines

Proved key properties related to hyperbolic splines

Enhanced understanding of solutions to the differential equation on real line

Abstract

In this note we discuss solutions of differential equation $(D^2-\alpha^2)^{k}u=0$ on $\mathbb{R}\setminus\mathbb{Z}$, which we call hyperbolic splines. We develop the fundamental function of interpolation and prove various properties related to these splines.