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Mat. Sb., 2000, Volume 191, Number 1, Pages 103–126 (Mi msb449)  

This article is cited in 29 scientific papers (total in 30 papers)

On everywhere divergence of trigonometric Fourier series

S. V. Konyagin

M. V. Lomonosov Moscow State University

Abstract: The following theorem is established.
Theorem. {\it Let a function $\varphi\colon[0,+\infty)\to[0,+\infty)$ and a sequence $\{\psi(m)\}$ satisfy the following condition: the function $\varphi(u)/u$ is non-decreasing on $(0,+\infty)$, $\psi(m)\geqslant 1$ $(m=1,2,…)$ and $\varphi(m)\psi(m)=o(m\sqrt{\ln m}/\sqrt{\ln\ln m} )$ as $m\to\infty$. Then there is a function $f\in L[-\pi,\pi]$ such that
$$ \int _{-\pi}^\pi\varphi(|f(x)|) dx<\infty $$
and $\limsup_{m\to\infty}S_m(f,x)/\psi(m)=\infty$ for all $x\in[-\pi,\pi]$ here $S_m(f)$ is the $m$-th partial sum of the trigonometric Fourier series of $f$}.


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English version:
Sbornik: Mathematics, 2000, 191:1, 97–120

Bibliographic databases:

UDC: 517.518.45
MSC: 42a20
Received: 11.06.1999

Citation: S. V. Konyagin, “On everywhere divergence of trigonometric Fourier series”, Mat. Sb., 191:1 (2000), 103–126; Sb. Math., 191:1 (2000), 97–120

Citation in format AMSBIB
\by S.~V.~Konyagin
\paper On everywhere divergence of trigonometric Fourier series
\jour Mat. Sb.
\yr 2000
\vol 191
\issue 1
\pages 103--126
\jour Sb. Math.
\yr 2000
\vol 191
\issue 1
\pages 97--120

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    This publication is cited in the following articles:
    1. Oniani G., “On Sjolin-Soria-Antonov Type Extrapolation For Locally Compact Groups and a.E. Convergence of Vilenkin-Fourier Series”, Acta Math. Hung.  crossref  mathscinet  isi
    2. Arias-De-Reyna J., “Pointwise convergence of Fourier series”, J. London Math. Soc. (2), 65 (2002), 139–153  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    3. N. Yu. Antonov, “Almost everywhere convergence over cubes of multiple trigonometric Fourier series”, Izv. Math., 68:2 (2004), 223–241  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. 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”, Math. Notes, 76:5 (2004), 606–619  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. Lacey M.T., “Carleson's theorem: proof, complements, variations”, Publ. Mat., 48:2 (2004), 251–307  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    6. N. Yu. Antonov, “Growth rate of sequences of multiple rectangular Fourier sums”, Proc. Steklov Inst. Math. (Suppl.), 2005no. , suppl. 2, S9–S29  mathnet  mathscinet  zmath  elib
    7. S. V. Konyagin, “Divergence everywhere of subsequences of partial sums of trigonometric Fourier series”, Proc. Steklov Inst. Math. (Suppl.), 2005no. , suppl. 2, S167–S175  mathnet  mathscinet  zmath  elib
    8. G. A. Karagulian, “Everywhere Divergent $\Phi$-Means of Fourier Series”, Math. Notes, 80:1 (2006), 47–56  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. N. Yu. Antonov, “On the almost everywhere convergence of sequences of multiple rectangular Fourier sums”, Proc. Steklov Inst. Math. (Suppl.), 264, suppl. 1 (2009), S1–S18  mathnet  crossref  isi  elib
    10. 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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    11. N. Yu. Antonov, “On the growth rate of arbitrary sequences of double rectangular Fourier sums”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S14–S20  mathnet  crossref  isi  elib
    12. Bochkarev S.V., “On a Problem of Hardy for Walsh-Fourier Series”, Doklady Mathematics, 81:3 (2010), 390–391  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    13. I. V. Polyakov, “Example of a Divergent Fourier Series in the Vilenkin System”, Math. Notes, 89:5 (2011), 734–740  mathnet  crossref  crossref  mathscinet  isi
    14. I. V. Polyakov, “Examples of divergent Fourier series for a wide class of rearranged Walsh–Paley system”, Moscow University Mathematics Bulletin, 68:1 (2013), 1–6  mathnet  crossref
    15. G. Gát, U. Goginava, G. Karagulyan, “On everywhere divergence of the strong Φ-means of Walsh–Fourier series”, Journal of Mathematical Analysis and Applications, 2014  crossref  mathscinet  scopus  scopus  scopus
    16. I. V. Polyakov, “Estimates of the Dirichlet Kernel and Divergent Fourier Series in the Walsh–Kaczmarz System”, Math. Notes, 95:2 (2014), 234–246  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    17. R. M. Trigub, “Summability of trigonometric Fourier series at $d$-points and a generalization of the Abel–Poisson method”, Izv. Math., 79:4 (2015), 838–858  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. N. Yu. Antonov, “On almost everywhere convergence for lacunary sequences of multiple rectangular Fourier sums”, Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 43–59  mathnet  crossref  mathscinet  isi  elib
    19. R. M. Trigub, “Almost Everywhere Summability of Fourier Series with Indication of the Set of Convergence”, Math. Notes, 100:1 (2016), 139–153  mathnet  crossref  crossref  mathscinet  isi  elib
    20. Lie V., “Pointwise Convergence of Fourier Series (i). on a Conjecture of Konyagin”, J. Eur. Math. Soc., 19:6 (2017), 1655–1728  crossref  mathscinet  zmath  isi  scopus
    21. Nikolai Yu. Antonov, “On $\Lambda$-convergence almost everywhere of multiple trigonometric Fourier series”, Ural Math. J., 3:2 (2017), 14–21  mathnet  crossref  mathscinet
    22. Weisz F., “Convergence and Summability of Fourier Transforms and Hardy Spaces”, Convergence and Summability of Fourier Transforms and Hardy Spaces, Applied and Numerical Harmonic Analysis, Birkhauser Boston, 2017, 1–435  crossref  mathscinet  isi
    23. Mastylo M., Rodriguez-Piazza L., “Convergence Almost Everywhere of Multiple Fourier Series Over Cubes”, Trans. Am. Math. Soc., 370:3 (2018), 1629–1659  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    24. S. V. Bochkarev, “An abstract Kolmogorov theorem, and an application to metric spaces and topological groups”, Sb. Math., 209:11 (2018), 1575–1602  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    25. B. S. Kashin, Yu. V. Malykhin, V. Yu. Protasov, K. S. Ryutin, I. D. Shkredov, “Sergei Vladimirovich Konyagin turns 60”, Proc. Steklov Inst. Math., 303 (2018), 1–9  mathnet  crossref  crossref  mathscinet  isi  elib
    26. Edmunds D., Gogatishvili A., Kopaliani T., “Construction of Function Spaces Close to l With Associate Space Close to l-1”, J. Fourier Anal. Appl., 24:6 (2018), 1539–1553  crossref  mathscinet  zmath  isi  scopus
    27. Lie V., “The Pointwise Convergence of Fourier Series (II). Strong l(1)Case For the Lacunary Carleson Operator”, Adv. Math., 357 (2019), 106831  crossref  mathscinet  zmath  isi
    28. Singh P., “Novel Generalized Fourier Representations and Phase Transforms”, Digit. Signal Prog., 106 (2020), 102830  crossref  isi
    29. Getsadze R., “On the Divergence of Double Fourier-Walsh-Paley Series of Continuous Functions”, Acta Sci. Math., 86:1-2 (2020), 287–302  crossref  mathscinet  isi
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