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 Mat. Zametki, 1969, Volume 6, Issue 1, Pages 47–54 (Mi mz6896)

On the best approximation in the metric of $L$ to certain classes of functions by Haar-system polynomials
Abstract: Let $H_\omega$, $H_\omega^L$ be classes of functions $f(x)$ whose modulus of continuity $\omega(f;t)$ and, respectively, integral modulus of continuity $\omega(f;t)_L$ do not exceed a given modulus of continuity \omega(t)$, while$H_V$is a class of functions$f(x)$whose variation$\mathop V\limits_0^1f$fdoes not exceed a given number$V>0$. Bounds are obtained for the upper limit of the best approximations in the metric of$L$by Haar-system polynomials on the classes just introduced (on the class$H_\omega^L$only when$\omega(t)=Kt$). These bounds are exact for class$H_V$and, in case$\omega(t)$is convex, also for the classes$H_\omega$and$H\omega^L$. Full text: PDF file (403 kB) English version: Mathematical Notes, 1969, 6:1, 487–491 Bibliographic databases: UDC: 517.5 Received: 05.08.1968 Citation: N. P. Khoroshko, “On the best approximation in the metric of$L$to certain classes of functions by Haar-system polynomials”, Mat. Zametki, 6:1 (1969), 47–54; Math. Notes, 6:1 (1969), 487–491 Citation in format AMSBIB \Bibitem{Kho69} \by N.~P.~Khoroshko \paper On the best approximation in the metric of$L$to certain classes of functions by Haar-system polynomials \jour Mat. Zametki \yr 1969 \vol 6 \issue 1 \pages 47--54 \mathnet{http://mi.mathnet.ru/mz6896} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=249895} \zmath{https://zbmath.org/?q=an:0188.14002|0179.37201} \transl \jour Math. Notes \yr 1969 \vol 6 \issue 1 \pages 487--491 \crossref{https://doi.org/10.1007/BF01450251}  Linking options: • http://mi.mathnet.ru/eng/mz6896 • http://mi.mathnet.ru/eng/mz/v6/i1/p47  SHARE: Citing articles on Google Scholar: Russian citations, English citations Related articles on Google Scholar: Russian articles, English articles This publication is cited in the following articles: 1. S. B. Vakarchuk, A. N. Shchitov, “Estimates for the error of approximation of classes of differentiable functions by Faber–Schauder partial sums”, Sb. Math., 197:3 (2006), 303–314 2. S. B. Vakarchuk, A. N. Shchitov, “Estimates for the error of approximation of functions in$L_p^1\$ by polynomials and partial sums of series in the Haar and Faber–Schauder systems”, Izv. Math., 79:2 (2015), 257–287