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Mat. Zametki, 1976, Volume 19, Issue 2, Pages 165–178 (Mi mz7736)  

This article is cited in 3 scientific papers (total in 3 papers)

A necessary condition for convergence of interpolating parabolic and cubic splines

N. L. Zmatrakov

Institute of Mathematics of the Ural Scientific Center of the USSR Academy of Sciences

Abstract: Let the sequence of nets $\Delta_n$ be such that $\lim\limits_{n\to\infty}\max\limits_ih_i^{(n)}=0$, where $h_i^{(n)}$ are the lengths of the segments of a net. The bound $\max\limits_{|i-j|=1}\frac{h_i^{(n)}}{h_j^{(n)}1-\alpha}\le R<\infty$ is necessary in order that interpolating parabolic and cubic splines converge for any function in $C(\alpha=0)$ or $C_\alpha(0<\alpha<1)$, where $C_\alpha$ is the class of functions satisfying a Lipschitz condition of order $\alpha$. It is also shown that this bound cannot essentially be weakened.

Full text: PDF file (632 kB)

English version:
Mathematical Notes, 1976, 19:2, 100–107

Bibliographic databases:

UDC: 517.5
Received: 10.03.1975

Citation: N. L. Zmatrakov, “A necessary condition for convergence of interpolating parabolic and cubic splines”, Mat. Zametki, 19:2 (1976), 165–178; Math. Notes, 19:2 (1976), 100–107

Citation in format AMSBIB
\Bibitem{Zma76}
\by N.~L.~Zmatrakov
\paper A~necessary condition for convergence of interpolating parabolic and cubic splines
\jour Mat. Zametki
\yr 1976
\vol 19
\issue 2
\pages 165--178
\mathnet{http://mi.mathnet.ru/mz7736}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=463759}
\zmath{https://zbmath.org/?q=an:0332.41007|0325.41006}
\transl
\jour Math. Notes
\yr 1976
\vol 19
\issue 2
\pages 100--107
\crossref{https://doi.org/10.1007/BF01098740}


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    Erratum

    This publication is cited in the following articles:
    1. Yu. S. Volkov, Yu. N. Subbotin, “50 years to Schoenberg's problem on the convergence of spline interpolation”, Proc. Steklov Inst. Math. (Suppl.), 288, suppl. 1 (2015), 222–237  mathnet  crossref  mathscinet  isi  elib
    2. I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “Cubic spline interpolation of functions with high gradients in boundary layers”, Comput. Math. Math. Phys., 57:1 (2017), 7–25  mathnet  crossref  crossref  isi  elib
    3. I. A. Blatov, A. I. Zadorin, E. V. Kitaeva, “Parabolic spline interpolation for functions with large gradient in the boundary layer”, Siberian Math. J., 58:4 (2017), 578–590  mathnet  crossref  crossref  isi  elib  elib
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