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 Mat. Zametki, 2004, Volume 76, Issue 5, Pages 651–665 (Mi mz136)

Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity

N. Yu. Antonov

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: We construct an example of a function from the class $H_1^{\omega^*}$ , where $\omega^*(t)=\sqrt{\log\log(t^{-1})/\log(t^{-1})}$, $0<t\le t_0$, whose trigonometric Fourier series is divergent almost everywhere. We obtain sharp integrability conditions for the majorants of the partial sums of trigonometric Fourier series in terms of whether the functions in question belong to the classes $H_1^\omega$.

DOI: https://doi.org/10.4213/mzm136

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English version:
Mathematical Notes, 2004, 76:5, 606–619

Bibliographic databases:

UDC: 517.518

Citation: N. Yu. Antonov, “Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity”, Mat. Zametki, 76:5 (2004), 651–665; Math. Notes, 76:5 (2004), 606–619

Citation in format AMSBIB
\Bibitem{Ant04} \by N.~Yu.~Antonov \paper Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity \jour Mat. Zametki \yr 2004 \vol 76 \issue 5 \pages 651--665 \mathnet{http://mi.mathnet.ru/mz136} \crossref{https://doi.org/10.4213/mzm136} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2129332} \zmath{https://zbmath.org/?q=an:1072.42003} \transl \jour Math. Notes \yr 2004 \vol 76 \issue 5 \pages 606--619 \crossref{https://doi.org/10.1023/B:MATN.0000049660.29081.bc} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000226356700002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-10344252851} 

• http://mi.mathnet.ru/eng/mz136
• https://doi.org/10.4213/mzm136
• http://mi.mathnet.ru/eng/mz/v76/i5/p651

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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. N. Yu. Antonov, “Almost Everywhere Divergent Subsequences of Fourier Sums of Functions from $\varphi(L)\cap H_1^\omega$”, Math. Notes, 85:4 (2009), 484–495
2. S. V. Konyagin, “Almost everywhere divergence of lacunary subsequences of partial sums of Fourier series”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S99–S106
3. N. Yu. Antonov, “Estimates for the growth order of sequences of multiple rectangular Fourier sums of integrable functions”, J. Math. Sci., 209:1 (2015), 1–11
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