A note concerning polyhyperbolic and related splines

TL;DR

This paper investigates the properties of two families of parametrized splines related to polynomial spline interpolation, focusing on their uniqueness and convergence rates.

Contribution

It establishes fundamental results on the uniqueness and convergence behavior of specific hyperbolic and related splines used in interpolation.

Findings

Proves uniqueness of the spline solutions

Establishes basic convergence rates for the splines

Analyzes the behavior of hyperbolic and related spline families

Abstract

This note concerns the interpolation problem with two parametrized families of splines related to polynomial spline interpolation. We address the questions of uniqueness and establish basic convergence rates for splines of the form $ s_\alpha = p\cosh(\alpha\cdot)+q\sinh(\alpha \cdot)$ and $t_\alpha = p+q\tanh(\alpha \cdot) $ between the nodes where $p,q\in\Pi_{k-1}$.