A note on some bounds between cubic spline interpolants depending on the boundary conditions: Application to a monotonicity property

TL;DR

This work analyzes whether cubic spline monotonicity can be achieved by adjusting only endpoint derivatives. By deriving a bound on how far two cubic Hermite splines can differ when their endpoint slopes change, the authors show that the influence of boundary data decays away from the interval ends. They then prove that for odd $n$ and monotone data, there exists an interior point where the spline overshoots the data, and this overshoot cannot be eliminated by modifying only the endpoint derivatives, meaning monotonicity cannot be guaranteed solely through boundary conditions. The result informs monotonicity-preserving strategies, indicating that interior derivative control or nonlinear interior reconstructions are required rather than relying on endpoint data alone.

Abstract

In the context of cubic splines, the authors have contributed to a recent paper dealing with the computation of nonlinear derivatives at the interior nodes so that monotonicity is enforced while keeping the order of approximation of the spline as high as possible. During the review process of that paper, one of the reviewers raised the question of whether a cubic spline interpolating monotone data could be forced to preserve monotonicity by imposing suitable values of the first derivative at the endpoints. Albeit a negative answer appears to be intuitive, we have found no results regarding this fact. In this short work we prove that the answer to that question is actually negative.

Theorems & Definitions (8)

• Lemma 2.1
• Proposition 3.1
• proof
• Corollary 3.2
• proof
• Theorem 4.1
• Proposition 4.2
• proof