Paper Info
Title
A Note on Modified Hermite Interpolation
Abstract
We discuss the problem of fitting a smooth regular curve $$\gamma {:}[0,T]{\rightarrow }\mathbb {E}^n$$ based on reduced data$$Q_m = \{q_i\}_{i = 0}^m$$ in arbitrary Euclidean space $$\mathbb {E}^n$$. The respective interpolation knots $${\mathcal T} = \{t_i\}_{i = 0}^m$$ satisfying $$q_i = \gamma (t_i)$$ are assumed to be unknown. In our setting the substitutes $${\mathcal T}_{\lambda }=\{{\hat{t}}_i\}_{i = 0}^m$$ of $${{\mathcal {T}}}$$ are selected according to the so-called exponential parameterization governed by $$Q_m$$ and $$\lambda \in [0,1]$$. A modified Hermite interpolant $$\hat{\gamma }^H$$ introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used here to fit $$(\hat{{\mathcal {T}}}_{\lambda },Q_m)$$. The case of $$\lambda = 1$$ (i.e. for cumulative chords) for general class of admissible samplings yields a sharp quartic convergence order in estimating $$\gamma {\in } C^4$$ by $${\hat{\gamma }}^H$$ [see Kozera (Stud Inf 25(4B–61):1–140, 2004) and Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004)]. As recently shown in Kozera and Wilkołazka (Math Comput Sci, 2018. https://doi.org/10.1007/s11786-018-0362-4) the remaining $$\lambda \in [0,1)$$ render a linear convergence order in $${\hat{\gamma }}^H\approx \gamma $$ for any $$Q_m$$ sampled more-or-less uniformly. The related analysis relies on comparing the difference $$\gamma -{\hat{\gamma }}^H\circ \phi ^H$$ in which $$\phi ^H$$ forms a special mapping between [0, T] and $$[0,{\hat{T}}]$$ with $${\hat{T}} = {\hat{t}}_m$$. In this paper: (a) several sufficient conditions enforcing $$\phi ^H$$ to yield a genuine reparameterization are first formulated and then analytically and symbolically simplified. The latter covers also the asymptotic case expressed in a simple form. Ultimately, the reformulated conditions can be algebraically verified and/or geometrically visualized, (b) additionally in Sect. 3, the sharpness of the asymptotics of $$\gamma -{\hat{\gamma }}^H\circ \phi ^H$$ [from Kozera and Wilkołazka (Math Comput Sci, 2018. https://doi.org/10.1007/s11786-018-0362-4)] is proved upon applying symbolic and analytic calculations in Mathematica.
Year
DOI
Venue
2020
10.1007/s11786-019-00434-3
Mathematics in Computer Science
Keywords
DocType
Volume
Interpolation, Reduced data, Sharpness in convergence order, Reparameterization, 65D05, 65D10, 65D18
Journal
14
Issue
ISSN
Citations 
2
1661-8270
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Ryszard Kozera116326.54
M. Wilkołazka210.72