Conjecture of error boundedness in a new Hermite interpolation problem via splines of odd-degree

TL;DR

This paper introduces a new Hermite spline interpolation problem of odd degree, conjectures bounded error independent of knot placement, and provides numerical evidence supporting the conjecture for functions with high smoothness.

Contribution

It proposes a novel odd-degree spline interpolation problem, formulates a conjecture on error bounds, and supports it with extensive numerical simulations.

Findings

Numerical evidence supports the conjecture for degrees 2k-1 with k=3,...,10.

The worst error occurs at the perfect spline with the same knots.

The problem is relevant for nonparametric density estimation methods.

Abstract

We present a Hermite interpolation problem via splines of odd-degree which, to the best knowledge of the authors, has not been considered in the literature on interpolation via odd-degree splines. In this new interpolation problem, we conjecture that the interpolation error is bounded in the supremum norm independently of the locations of the knots. Given an integer k greater than or equal to 3, our spline interpolant is of degree 2k-1 and with 2k-4 (interior) knots. Simulations were performed to check the validity of the conjecture. We present strong numerical evidence in support of the conjecture for k=3, ..., 10 when the interpolated function belongs to C^{(2k)}[0,1], the class of 2k-times continuously differentiable functions on [0,1]. In this case, the worst interpolation error is proved to be attained by the perfect spline of degree 2k with the same knots as the spline interpolant. This interpolation problem arises naturally in nonparametric estimation of a multiply monotone density via Least Squares and Maximum Likehood methods.