### Abstract

This work discusses the problem of fitting a regular curve (Formula presented.) based on reduced data points(Formula presented.) in arbitrary Euclidean space. The corresponding interpolation knots (Formula presented.) are assumed to be unknown. In this paper the missing knots are estimated by (Formula presented.) 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 (Formula presented.). In order to fit (Formula presented.), a modified Hermite interpolant (Formula presented.) (a (Formula presented.) piecewise-cubic) introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used. The sharp quartic convergence order for estimating (Formula presented.) by (Formula presented.) is proved in Kozera (Stud Inf 25(4B-61):1–140, 2004) and Kozera and Noakes (2004) only for (Formula presented.) 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 (Formula presented.). A slower linear convergence order in trajectory estimation is established for any (Formula presented.) 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 (Formula presented.) and based on (Formula presented.) together with some relevant applications are also briefly recalled in this paper.

Original language | English |
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Pages (from-to) | 1-13 |

Number of pages | 13 |

Journal | Mathematics in Computer Science |

DOIs | |

Publication status | E-pub ahead of print - 6 Jul 2018 |

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*Mathematics in Computer Science*, 1-13. https://doi.org/10.1007/s11786-018-0362-4

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*Mathematics in Computer Science*, pp. 1-13. https://doi.org/10.1007/s11786-018-0362-4

**A Modified Hermite Interpolation with Exponential Parameterization.** / Kozera, R.; Wilkołazka, M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A Modified Hermite Interpolation with Exponential Parameterization

AU - Kozera, R.

AU - Wilkołazka, M.

PY - 2018/7/6

Y1 - 2018/7/6

N2 - This work discusses the problem of fitting a regular curve (Formula presented.) based on reduced data points(Formula presented.) in arbitrary Euclidean space. The corresponding interpolation knots (Formula presented.) are assumed to be unknown. In this paper the missing knots are estimated by (Formula presented.) 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 (Formula presented.). In order to fit (Formula presented.), a modified Hermite interpolant (Formula presented.) (a (Formula presented.) piecewise-cubic) introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used. The sharp quartic convergence order for estimating (Formula presented.) by (Formula presented.) is proved in Kozera (Stud Inf 25(4B-61):1–140, 2004) and Kozera and Noakes (2004) only for (Formula presented.) 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 (Formula presented.). A slower linear convergence order in trajectory estimation is established for any (Formula presented.) 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 (Formula presented.) and based on (Formula presented.) together with some relevant applications are also briefly recalled in this paper.

AB - This work discusses the problem of fitting a regular curve (Formula presented.) based on reduced data points(Formula presented.) in arbitrary Euclidean space. The corresponding interpolation knots (Formula presented.) are assumed to be unknown. In this paper the missing knots are estimated by (Formula presented.) 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 (Formula presented.). In order to fit (Formula presented.), a modified Hermite interpolant (Formula presented.) (a (Formula presented.) piecewise-cubic) introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004) is used. The sharp quartic convergence order for estimating (Formula presented.) by (Formula presented.) is proved in Kozera (Stud Inf 25(4B-61):1–140, 2004) and Kozera and Noakes (2004) only for (Formula presented.) 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 (Formula presented.). A slower linear convergence order in trajectory estimation is established for any (Formula presented.) 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 (Formula presented.) and based on (Formula presented.) together with some relevant applications are also briefly recalled in this paper.

KW - Convergence order

KW - Interpolation

KW - Numerical analysis

KW - Reduced data

UR - http://www.scopus.com/inward/record.url?scp=85049602356&partnerID=8YFLogxK

U2 - 10.1007/s11786-018-0362-4

DO - 10.1007/s11786-018-0362-4

M3 - Article

SP - 1

EP - 13

JO - Mathematics in Computer Science

JF - Mathematics in Computer Science

SN - 1661-8270

ER -