Abstract
This work discusses the problem of fitting a regular curve \(\gamma \) based on reduced data points\(Q_m=(q_0,q_1,\dots ,q_m)\) in arbitrary Euclidean space. The corresponding interpolation knots \({\mathcal T}=(t_0,t_1,\dots ,t_m)\) are assumed to be unknown. In this paper the missing knots are estimated by \({\mathcal T}_{\lambda }=(\hat{t}_0,\hat{t}_1,\dots ,\hat{t}_m)\) in accordance with the so-called exponential parameterization (see Kvasov in Methods of shape-preserving spline approximation, World Scientific Publishing Company, Singapore, 2000) controlled by a single parameter \(\lambda \in [0,1]\). In order to fit \((\hat{\mathcal T}_{\lambda },Q_m)\), a modified Hermite interpolant \(\hat{\gamma }^H\) (a \(C^1\) piecewise-cubic) introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used. The sharp quartic convergence order for estimating \(\gamma \in C^4\) by \(\hat{\gamma }^H\) is proved in Kozera (Stud Inf 25(4B-61):1–140, 2004) and Kozera and Noakes (2004) only for \(\lambda =1\) and within the general class of admissible samplings. The main result of this paper extends the latter to the remaining cases of exponential parameterization covering all \(\lambda \in [0,1)\). A slower linear convergence order in trajectory estimation is established for any \(\lambda \in [0,1)\) and arbitrary more-or-less uniform sampling. The numerical tests conducted in Mathematica indicate the sharpness of the latter and confirm the necessity of more-or-less uniformity. Other interpolation schemes used to fit reduced data \(Q_m\) and based on \({\mathcal T}_{\lambda }\) together with some relevant applications are also briefly recalled in this paper.
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Kozera, R., Wilkołazka, M. A Modified Hermite Interpolation with Exponential Parameterization. Math.Comput.Sci. 13, 143–155 (2019). https://doi.org/10.1007/s11786-018-0362-4
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DOI: https://doi.org/10.1007/s11786-018-0362-4