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 Trudy Inst. Mat. i Mekh. UrO RAN, 2019, Volume 25, Number 2, Pages 67–74 (Mi timm1624)

Convergence of quartic interpolation splines

Yu. S. Volkovab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University

Abstract: The problem of interpolation by quartic splines according to Marsden's scheme is considered. It is shown that the calculation of an interpolating spline in terms of the coefficients of expansion of its second derivative in $L_1$-normalized quadratic B-splines yields a system of linear equations for the chosen parameters. The matrix of the system is pentadiagonal and has a column diagonal dominance, which makes it possible to efficiently calculate the required parameters and establish the convergence of the spline interpolation process according to Marsden's scheme for any function from the class $C^1$ on an arbitrary sequence of grids without any constraints. In Marsden's scheme, it is assumed that a knot grid is given and the interpolation nodes are chosen strictly in the middle. The established results are transferred to the case of interpolation by quartic splines according to Subbotin's scheme (the node grid and knot grid are swapped). Here the system of equations for the coefficients of expansion of the third derivative in $L_\infty$-normalized B-splines has a diagonal dominance, and the interpolation process converges for any interpolated function from the class $C^3$.

Keywords: quartic splines, interpolation, convergence, diagonally dominant matrices.

 Funding Agency Grant Number Ministry of Science and Higher Education of the Russian Federation 0314-2016-0013 Russian Foundation for Basic Research 19-51-12008 This work was supported by the Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences (state contract no. 0314-2016-0013), and partially by the Russian Foundation for Basic Research and the German Research Foundation (project no. 19-51-12008).

DOI: https://doi.org/10.21538/0134-4889-2019-25-2-67-74

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UDC: 519.65
MSC: 41A05, 41A15, 41A25

Citation: Yu. S. Volkov, “Convergence of quartic interpolation splines”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 2, 2019, 67–74

Citation in format AMSBIB
\Bibitem{Vol19} \by Yu.~S.~Volkov \paper Convergence of quartic interpolation splines \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2019 \vol 25 \issue 2 \pages 67--74 \mathnet{http://mi.mathnet.ru/timm1624} \crossref{https://doi.org/10.21538/0134-4889-2019-25-2-67-74} \elib{http://elibrary.ru/item.asp?id=38071601}