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 Izv. Akad. Nauk SSSR Ser. Mat., 1977, Volume 41, Issue 4, Pages 937–958 (Mi izv1874)

On $(H,k)$-summability of multiple trigonometric Fourier series

Abstract: A theorem is proved from which, in particular, it follows that if $f\in L(\ln^+L)^{N-1}$ on $T^N\equiv[-\pi,\pi]^N$, then the multiple trigonometric Fourier series of $f$ and all conjugate series are $(H,k)$-summable almost everywhere on $T^N$ for every $k>0$.
In the case where $f\in L(\ln^+L)^{N+1}$ this result was obtained by Marcinkiewicz (Collected papers, PWN, Warsaw, 1964).
That it is unimprovable, in a certain sense, follows from a result of Saks (On the strong derivatives of functions of intervals, Fund. Math. 25 (1935), 235–252).
Bibliography: 15 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1977, 11:4, 889–908

Bibliographic databases:

UDC: 517.5
MSC: 42A24, 42A92, 42A40, 40G05

Citation: L. D. Gogoladze, “On $(H,k)$-summability of multiple trigonometric Fourier series”, Izv. Akad. Nauk SSSR Ser. Mat., 41:4 (1977), 937–958; Math. USSR-Izv., 11:4 (1977), 889–908

Citation in format AMSBIB
\Bibitem{Gog77} \by L.~D.~Gogoladze \paper On $(H,k)$-summability of multiple trigonometric Fourier series \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1977 \vol 41 \issue 4 \pages 937--958 \mathnet{http://mi.mathnet.ru/izv1874} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=473711} \zmath{https://zbmath.org/?q=an:0362.42003} \transl \jour Math. USSR-Izv. \yr 1977 \vol 11 \issue 4 \pages 889--908 \crossref{https://doi.org/10.1070/IM1977v011n04ABEH001750} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. L. D. Gogoladze, “On strong summability almost everywhere”, Math. USSR-Sb., 63:1 (1989), 153–164
2. M. I. Dyachenko, “Some problems in the theory of multiple trigonometric series”, Russian Math. Surveys, 47:5 (1992), 103–171
3. V. A. Rodin, “The tensor BMO-property of the sequence of partial sums of a multiple Fourier series”, Russian Acad. Sci. Sb. Math., 80:1 (1995), 211–224
4. 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
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