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 Mat. Sb. (N.S.), 1975, Volume 96(138), Number 2, Pages 189–211 (Mi msb3123)

On convergence of Riesz spherical means of multiple Fourier series

B. I. Golubov

Abstract: An $N$-dimensional analog is proved of a theorem of Plessner and Ul'yanov on equivalent conditions for convergence of certain series and integrals. There is obtained from it a sufficient condition on the quadratic modulus of continuity of a periodic function of $N\geqslant2$ variables ensuring the a.e. convergence of the spherical sums of its Fourier series. A two-dimensional analog of a theorem of Luzin and Denjoy and an $N$-dimensional analog of the Dini–Lipschitz criterion are proved. A necessary and sufficient condition on a function $\Phi(u)$ is derived ensuring the pointwise convergence of the Riesz spherical means of critical order of multiple Fourier series of functions of bounded $\Phi$-variation.
Bibliography: 33 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 25:2, 177–197

Bibliographic databases:

UDC: 517.522.3
MSC: Primary 42A20, 42A92; Secondary 42A28, 26A15, 26A45

Citation: B. I. Golubov, “On convergence of Riesz spherical means of multiple Fourier series”, Mat. Sb. (N.S.), 96(138):2 (1975), 189–211; Math. USSR-Sb., 25:2 (1975), 177–197

Citation in format AMSBIB
\Bibitem{Gol75} \by B.~I.~Golubov \paper On convergence of Riesz spherical means of multiple Fourier series \jour Mat. Sb. (N.S.) \yr 1975 \vol 96(138) \issue 2 \pages 189--211 \mathnet{http://mi.mathnet.ru/msb3123} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=382993} \zmath{https://zbmath.org/?q=an:0305.42022} \transl \jour Math. USSR-Sb. \yr 1975 \vol 25 \issue 2 \pages 177--197 \crossref{https://doi.org/10.1070/SM1975v025n02ABEH002205} 

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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. B. I. Golubov, “On the summability of Fourier integrals by Riesz spherical means”, Math. USSR-Sb., 33:4 (1977), 501–518
2. Christopher Meaney, “On almost-everywhere convergent eigenfunction expansions of the Laplace–Beltrami operator”, Math Proc Camb Phil Soc, 92:1 (1982), 129
3. M. I. Dyachenko, “Some problems in the theory of multiple trigonometric series”, Russian Math. Surveys, 47:5 (1992), 103–171
4. M. I. Dyachenko, “$U$-convergence almost everywhere of double Fourier series”, Sb. Math., 186:1 (1995), 47–64
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