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Mat. Zametki, 1970, Volume 7, Issue 1, Pages 31–42 (Mi mz6990)  

This article is cited in 9 scientific papers (total in 10 papers)

Order of the best spline approximations of some classes of functions

Yu. N. Subbotin, N. I. Chernykh

V. A. Steklov Institute of Mathematics, Sverdlovsk Branch of the Academy of Sciences of USSR

Abstract: The rate of decrease of the upper bounds of the best spline approximations $E_{m,n}(f)_p$ with undetermined $n$ nodes in the metric of the space $L_p(0,1)$ $(1\le p\le\infty)$ is studied in a class of functions $f(x)$ for which $\|f^{(m+1)}(x)\|_{L_q(0,1)}\le1$ $(1\le q\le\infty)$ or $\mathrm{var}\{f^{(m)}(x);0,1\}\le1$ ($m=1,2,…$, the preceding derivative is assumed absolutely continuous). An exact order of decrease of the mentioned bounds is found as $n\to\infty$, and asymptotic formulas are obtained for $p=\infty$ and $1\le q\le\infty$ in the case of an approximation by broken lines $(m=1)$. The simultaneous approximation of the function and its derivatives by spline functions and their appropriate derivatives is also studied.

Full text: PDF file (776 kB)

English version:
Mathematical Notes, 1970, 7:1, 20–26

Bibliographic databases:

UDC: 517.5
Received: 05.05.1969

Citation: Yu. N. Subbotin, N. I. Chernykh, “Order of the best spline approximations of some classes of functions”, Mat. Zametki, 7:1 (1970), 31–42; Math. Notes, 7:1 (1970), 20–26

Citation in format AMSBIB
\Bibitem{SubChe70}
\by Yu.~N.~Subbotin, N.~I.~Chernykh
\paper Order of the best spline approximations of some classes of functions
\jour Mat. Zametki
\yr 1970
\vol 7
\issue 1
\pages 31--42
\mathnet{http://mi.mathnet.ru/mz6990}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=259439}
\zmath{https://zbmath.org/?q=an:0198.09001|0195.35103}
\transl
\jour Math. Notes
\yr 1970
\vol 7
\issue 1
\pages 20--26
\crossref{https://doi.org/10.1007/BF01093336}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Yu. N. Subbotin, “Inheritance of monotonicity and convexity in local approximations”, Comput. Math. Math. Phys., 33:7 (1993), 879–884  mathnet  mathscinet  zmath  isi
    2. K. V. Kostousov, V. T. Shevaldin, “Approximation by local trigonometric splines”, Math. Notes, 77:3 (2005), 326–334  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. V. T. Shevaldin, “Approksimatsiya lokalnymi parabolicheskimi splainami s proizvolnym raspolozheniem uzlov”, Sib. zhurn. vychisl. matem., 8:1 (2005), 77–88  mathnet  zmath
    4. E. V. Shevaldina, “Approksimatsiya lokalnymi eksponentsialnymi splainami s proizvolnymi uzlami”, Sib. zhurn. vychisl. matem., 9:4 (2006), 391–402  mathnet
    5. “Sovmestnaya nauchnaya deyatelnost Yu. N. Subbotina i N. I. Chernykh”, Tr. IMM UrO RAN, 17, no. 3, 2011, 4–7  mathnet
    6. Pakhnutov I.A., “Prodolzhenie setochnoi funktsii s zadannymi usloviyami”, Izvestiya kaliningradskogo gosudarstvennogo tekhnicheskogo universiteta, 2012, no. 26, 74–80  elib
    7. I. P. Irodova, “Piecewise polynomial approximation methods in the theory of Nikol'skiĭ–Besov spaces”, Journal of Mathematical Sciences, 209:3 (2015), 319–480  mathnet  crossref
    8. A. A. Vladimirov, I. A. Sheipak, “On the Neumann Problem for the Sturm–Liouville Equation with Cantor-Type Self-Similar Weight”, Funct. Anal. Appl., 47:4 (2013), 261–270  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. A.-R. K. Ramazanov, V. G. Magomedova, “O nailuchshikh priblizheniyakh nepreryvno differentsiruemykh funktsii splainami po dvukhtochechnym ratsionalnym interpolyantam”, Dagestanskie elektronnye matematicheskie izvestiya, 2016, no. 5, 49–55  mathnet  crossref  elib
    10. A.-R. K. Ramazanov, V. G. Magomedova, “Otsenki skorosti skhodimosti splainov po trekhtochechnym ratsionalnym interpolyantam dlya nepreryvnykh i nepreryvno differentsiruemykh funktsii”, Tr. IMM UrO RAN, 23, no. 3, 2017, 224–233  mathnet  crossref  elib
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