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 Mat. Zametki, 2010, Volume 87, Issue 3, Pages 429–442 (Mi mz4508)

Approximation of Classes $B^r_{p,\theta}$ of Periodic Functions of One and Several Variables

A. S. Romanyuk

Institute of Mathematics, Ukrainian National Academy of Sciences

Abstract: We obtain order-sharp estimates of best approximations to the classes $B^r_{p,\theta}$ of periodic functions of several variables in the space $L_q$, $1\le p,q\le\infty$, by trigonometric polynomials with “numbers” of harmonics from step hyperbolic crosses. In the one-dimensional case, we establish the order of deviation of Fourier partial sums of functions from the classes $B^{r_1}_{1,\theta}$ in the space $L_1$.

Keywords: class $B^r_{p,\theta}$ of periodic functions, trigonometric polynomial, hyperbolic cross, Bernoulli kernel, Fourier hyperbolic sum, Valée-Poussin kernel, Fejér kernel

DOI: https://doi.org/10.4213/mzm4508

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English version:
Mathematical Notes, 2010, 87:3, 403–415

Bibliographic databases:

Document Type: Article
UDC: 517.51

Citation: A. S. Romanyuk, “Approximation of Classes $B^r_{p,\theta}$ of Periodic Functions of One and Several Variables”, Mat. Zametki, 87:3 (2010), 429–442; Math. Notes, 87:3 (2010), 403–415

Citation in format AMSBIB
\Bibitem{Rom10} \by A.~S.~Romanyuk \paper Approximation of Classes $B^r_{p,\theta}$ of Periodic Functions of One and Several Variables \jour Mat. Zametki \yr 2010 \vol 87 \issue 3 \pages 429--442 \mathnet{http://mi.mathnet.ru/mz4508} \crossref{https://doi.org/10.4213/mzm4508} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2761599} \zmath{https://zbmath.org/?q=an:05791063} \transl \jour Math. Notes \yr 2010 \vol 87 \issue 3 \pages 403--415 \crossref{https://doi.org/10.1134/S0001434610030120} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000279034600012} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77954014165} 

• http://mi.mathnet.ru/eng/mz4508
• https://doi.org/10.4213/mzm4508
• http://mi.mathnet.ru/eng/mz/v87/i3/p429

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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. Stasyuk S.A., “Best approximation of periodic functions of several variables from the classes $MB_{p,\theta}^\omega$”, Ukrainian Math. J., 64:1 (2012), 156–161
2. Myronyuk V.V., “Approximation of the Classes of Periodic Functions of Many Variables by Fourier Sums in the Space l (P) with P=1, a”, Ukr. Math. J., 64:9 (2013), 1370–1381
3. Romanyuk A.S., “Entropy Numbers and Widths For the Classes of Periodic Functions of Many Variables”, Ukr. Math. J., 68:10 (2017), 1620–1636
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