Cumulative chords,Piecewise-Quadratics and Piecewise-cubics

Length Estimation for Exponential Parameterization and ε-Uniform Samplings

“…5 define genuine re-parameterizations (see [14]). As already indicated in this paper, the latter is independently established for λ = 1 (in fact for general admissible samplings (2) -see [4]), and for λ = 0 (see [3]) and finally for uniform samplings (see [7]). …”

“…[3]) or λ = 1 (see e.g. [4] or [7]) were earlier studied in the context of examining the asymptotics for trajectory and length estimation. Recent results by [14] and [15] with full mathematical proofs guarantee sharp asymptotics for the trajectory estimation covering the missing cases of λ ∈ (0, 1).…”

“…5 guaranteeing ψ to be a re-parameterization (see [5]) are fulfilled. The latter is automatically satisfied by [14] for ε > 0 and λ ∈ [0, 1] or by [4] for λ = 1 and samplings (2)…”

“…incorporate the geometry of Q m and as such render much better asymptotic orders for length estimation (at least for r = 2, 3) as opposed to λ = 0 and (9) (see [4] and [7]). Indeed we have: …”

“…Finally, (11) yields accelerated convergence orders β ε (1) = min{4, 3 + ε}) which again coincide with non-reduced data case claimed by (7). The relevant numerical tests confirming the sharpness of (10) and (11) are conducted in [4] and [7].…”