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Izv. RAN. Ser. Mat., 2004, Volume 68, Issue 2, Pages 3–22 (Mi izv472)  

This article is cited in 7 scientific papers (total in 8 papers)

Almost everywhere convergence over cubes of multiple trigonometric Fourier series

N. Yu. Antonov


Abstract: Under certain conditions on a function $\varphi\colon[0,+\infty)\to[0,+\infty)$ we prove a theorem asserting that the convergence almost everywhere of trigonometric Fourier series for all functions of class $\varphi(L)_{[-\pi,\pi)}$ implies the convergence over cubes of the multiple Fourier series and all its conjugates for an arbitrary function $f\in\varphi(L)(\log^+L)^{d-1}_{[-\pi,\pi)^d}$, $d\in\mathbb N$. It follows from this and an earlier result of the author on the convergence almost everywhere of Fourier series of functions of one variable and class $L(\log^+L)(\log^+\log^+\log^+L)_{[-\pi,\pi)}$ that if $f\in L(\log^+L)^d(\log^+\log^+\log^+L)_{[-\pi,\pi)^d}$, $d\in\mathbb N$, then the Fourier series of $f$ and all its conjugates converge over cubes almost everywhere.

DOI: https://doi.org/10.4213/im472

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English version:
Izvestiya: Mathematics, 2004, 68:2, 223–241

Bibliographic databases:

UDC: 517.518
MSC: 41A30, 42A10, 42A20, 42A92, 42B05, 43A50, 46B15
Received: 24.11.2002

Citation: N. Yu. Antonov, “Almost everywhere convergence over cubes of multiple trigonometric Fourier series”, Izv. RAN. Ser. Mat., 68:2 (2004), 3–22; Izv. Math., 68:2 (2004), 223–241

Citation in format AMSBIB
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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, “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
    2. N. Yu. Antonov, “Note on estimates for the growth order of sequences of multiple rectangular Fourier sums”, Proc. Steklov Inst. Math. (Suppl.), 277, suppl. 1 (2012), 4–8  mathnet  crossref  isi  elib
    3. 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  elib
    4. Nikolai Yu. Antonov, “On $\Lambda$-convergence almost everywhere of multiple trigonometric Fourier series”, Ural Math. J., 3:2 (2017), 14–21  mathnet  crossref
    5. Goginava U., “Almost Everywhere Strong Summability of Cubic Partial Sums of D-Dimensional Walsh-Fourier Series”, Math. Inequal. Appl., 20:4 (2017), 1051–1066  crossref  mathscinet  zmath  isi  scopus
    6. 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
    7. 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
    8. 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  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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