### Abstract

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 E^{n}. 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 γ_{3}: [0, ] →T E^{n} 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 γ_{3} 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 γ_{3} 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.

Original language | English |
---|---|

Pages (from-to) | 72-94 |

Number of pages | 23 |

Journal | Mathematical Modelling and Analysis |

Volume | 24 |

Issue number | 1 |

DOIs | |

Publication status | Published - 21 Nov 2019 |

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*Mathematical Modelling and Analysis*,

*24*(1), 72-94. https://doi.org/10.3846/mma.2019.006

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*Mathematical Modelling and Analysis*, vol. 24, no. 1, pp. 72-94. https://doi.org/10.3846/mma.2019.006

**Convergence order in trajectory estimation by piecewise-cubics and exponential parameterization.** / Kozera, Ryszard; Wilkołlazka, Magdalena.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Convergence order in trajectory estimation by piecewise-cubics and exponential parameterization

AU - Kozera, Ryszard

AU - Wilkołlazka, Magdalena

PY - 2019/11/21

Y1 - 2019/11/21

N2 - This paper disusses the problem of estimating the trajetory of the unknown urve γ from the sequene of m + 1 interpolation points Qm = {γ(ti)}m i=0 in arbitrary Eulidean spae En. The respetive knots Tm = {ti}m i=0 (in asending order) are assumed to be unknown. Suh Qm isT oined reduced data. In our setting, a pieewise-ubi Lagrange interpolation γ3: [0, ] →T En is applied to fit Qm. Here, the missing knots Tm are replaed by their estimatesm = {ti}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 Qm. 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 Qm. 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 Qm 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.

AB - This paper disusses the problem of estimating the trajetory of the unknown urve γ from the sequene of m + 1 interpolation points Qm = {γ(ti)}m i=0 in arbitrary Eulidean spae En. The respetive knots Tm = {ti}m i=0 (in asending order) are assumed to be unknown. Suh Qm isT oined reduced data. In our setting, a pieewise-ubi Lagrange interpolation γ3: [0, ] →T En is applied to fit Qm. Here, the missing knots Tm are replaed by their estimatesm = {ti}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 Qm. 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 Qm. 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 Qm 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.

KW - Convergence order and sharpness

KW - Interpolation

KW - Reduced data

UR - http://www.scopus.com/inward/record.url?scp=85059703873&partnerID=8YFLogxK

U2 - 10.3846/mma.2019.006

DO - 10.3846/mma.2019.006

M3 - Article

VL - 24

SP - 72

EP - 94

JO - Mathematical Modelling and Analysis

JF - Mathematical Modelling and Analysis

SN - 1392-6292

IS - 1

ER -