关于通过奇次样条的新Hermite插值问题中误差有界性的猜想
Conjecture of error boundedness in a new Hermite interpolation problem via splines of odd-degree
AI总结：
提出一种新的奇次样条Hermite插值问题，猜想插值误差在最大模下与节点位置无关，并通过数值实验为k=3,...,10提供强证据。
摘要：
我们提出一个通过奇次样条的Hermite插值问题，据作者所知，这在关于奇次样条插值的文献中尚未被考虑。在这个新的插值问题中，我们猜想插值误差在最大模下与节点位置无关。给定整数k≥3，我们的样条插值函数次数为2k-1，且有2k-4个（内部）节点。进行模拟以检验猜想的有效性。我们提供强有力的数值证据支持k=3,...,10时的猜想，当被插值函数属于C^{(2k)}[0,1]（即[0,1]上2k次连续可微函数类）时。在这种情况下，证明最坏插值误差由具有与样条插值函数相同节点的2k次完美样条达到。该插值问题自然出现在通过最小二乘和最大似然方法对多重单调密度的非参数估计中。
英文摘要：
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.
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