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 Mat. Zametki, 2009, Volume 85, Issue 4, Pages 502–515 (Mi mz6640)

Almost Everywhere Divergent Subsequences of Fourier Sums of Functions from $\varphi(L)\cap H_1^\omega$

N. Yu. Antonov

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

Abstract: For a gap sequence of natural numbers $\{n_k\}^\infty_{k=1}$, for a nondecreasing function $\varphi\colon[0,+\infty)\to[0,+\infty)$ such that $\varphi(u)=o(u\ln\ln u)$ as $u\to\infty$, and a modulus of continuity satisfying the condition $(\ln k)^{-1}=O(\omega(n_k^{-1}))$, we present an example of a function $F\in\varphi(L)\cap H_1^\omega$ with an almost everywhere divergent subsequence $\{S_{n_k}(F,x)\}$ of the sequence of partial sums of the trigonometric Fourier series of the function $F$.

Keywords: Fourier sum, gap sequence, trigonometric Fourier series, modulus of continuity, Dirichlet kernel, Lebesgue measurability, Jensen's inequality

DOI: https://doi.org/10.4213/mzm6640

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English version:
Mathematical Notes, 2009, 85:4, 484–495

Bibliographic databases:

UDC: 517.518
Revised: 04.07.2008

Citation: N. Yu. Antonov, “Almost Everywhere Divergent Subsequences of Fourier Sums of Functions from $\varphi(L)\cap H_1^\omega$”, Mat. Zametki, 85:4 (2009), 502–515; Math. Notes, 85:4 (2009), 484–495

Citation in format AMSBIB
\Bibitem{Ant09} \by N.~Yu.~Antonov \paper Almost Everywhere Divergent Subsequences of Fourier Sums of Functions from $\varphi(L)\cap H_1^\omega$ \jour Mat. Zametki \yr 2009 \vol 85 \issue 4 \pages 502--515 \mathnet{http://mi.mathnet.ru/mz6640} \crossref{https://doi.org/10.4213/mzm6640} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2549413} \zmath{https://zbmath.org/?q=an:05628178} \elib{https://elibrary.ru/item.asp?id=15305483} \transl \jour Math. Notes \yr 2009 \vol 85 \issue 4 \pages 484--495 \crossref{https://doi.org/10.1134/S0001434609030201} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000266561100020} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-70249106439} 

• http://mi.mathnet.ru/eng/mz6640
• https://doi.org/10.4213/mzm6640
• http://mi.mathnet.ru/eng/mz/v85/i4/p502

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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. 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
2. Lie V., “Pointwise Convergence of Fourier Series (i). on a Conjecture of Konyagin”, J. Eur. Math. Soc., 19:6 (2017), 1655–1728
3. Lie V., “The Pointwise Convergence of Fourier Series (II). Strong l(1)Case For the Lacunary Carleson Operator”, Adv. Math., 357 (2019), 106831
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