Abstract
A class of rational quartic/cubic spline interpolants with three families of local control parameters is constructed, which can be \(C^2\) continuous without solving a global linear or non-linear system of consistency equations. The effects of the local control parameters on generating interpolation spline are illustrated. For \(C^2\) interpolation, the given interpolant can locally reproduce quadratic polynomials and has \(O(h^2)\) or \(O(h^3)\) convergence. Simple schemes for the proposed interpolant to preserve the shape of positive, monotonic, and/or convex set of data are derived. Sufficient conditions for the interpolant to lie strictly between two given piecewise linear functions are also given. Moreover, the important inflection property of the rational quartic/cubic interpolation spline is discussed in detail.
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Acknowledgements
The author is deeply grateful to the anonymous referees for their insightful comments and constructive suggestions for great improvement of this paper. The research is supported by the National Natural Science Foundation of China (Nos. 60970097, 11271376, 11771453, 61802129), the Postdoctoral Science Foundation of China (No. 2015M571931), and the Fundamental Research Funds for the Central Universities (No. 2017MS121).
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Zhu, Y. \(\varvec{C}^2\) Rational Quartic/Cubic Spline Interpolant with Shape Constraints. Results Math 73, 127 (2018). https://doi.org/10.1007/s00025-018-0883-9
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DOI: https://doi.org/10.1007/s00025-018-0883-9