A Modified Hermite Interpolation with Exponential Parameterization

Abstract

This work discusses the problem of fitting a regular curve gamma based on reduced data pointsQ_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.