Convergence order in trajectory estimation by piecewise-cubics and exponential parameterization

This paper disusses the problem of estimating the trajetory of the unknown urve γ from the sequene of m + 1 interpolation points Q_m = {γ(t_i)}^m _i=0 in arbitrary Eulidean spae Eⁿ. The respetive knots T_m = {t_i}^m _i=0 (in asending order) are assumed to be unknown. Suh Q_m isT oined reduced data. In our setting, a pieewise-ubi Lagrange interpolation γ₃: [0, ] →T Eⁿ is applied to fit Q_m. Here, the missing knots T_m are replaed by their estimates_m = {t_i}^m _i=0 in aordane with the exponential parameterization. The latter is ontrolled by a single parameter λ ∈ [0, 1]. This work analyzes the intrinsi asymptotis in approximating γ by γ₃ based on the exponential parameterization and Q_m. The multiple goals are ahieved. Firstly, the existing result established for λ = 1 (i.e. for the umulative hord parameterization) is extended to the remaining ases of λ ∈ [0, 1) and more-or-less uniformly sampled Q_m. As demonstrated herein, a quarti onvergene order α(1) = 4 in trajetory estimation drops disontinuously to the linear one α(λ) = 1, for all λ ∈ [0, 1). Seondly, the asymptotis derived in this paper is also analytially proved to be sharp with the aid of illustrative examples. Thirdly, the latter is verified in affirmative upon onduting numerial testing. Next, the neessity of more-or-less uniformity imposed on Q_m is shown to be indispensable. In addition, several suffiient onditions for γ₃ to be reparameterizable to the domain of γ are formulated. Lastly, the motivation for using the exponential parameterization with λ ∈ [0, 1) is also outlined.