A note concerning polyhyperbolic and related splines

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 $p,q\in\Pi_{k-1}$.