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Mat. Sb., 1993, Volume 184, Number 1, Pages 15–40 (Mi msb954)  

This article is cited in 1 scientific paper (total in 1 paper)

On the order of growth $o(\log\log n)$ of the partial sums of Fourier–Stieltjes series of random measures

G. A. Karagulian

Institute of Mathematics, National Academy of Sciences of Armenia

Abstract: Random measures of the form
$$ \sum_{i=1}^\infty m_i\delta_{\theta_i}, \qquad \sum_{i=1}^\infty|m_i|<\infty, $$
are considered, where $\delta_{\theta_i}$ is a unit mass concentrated at the point $\theta_i\in(0;2\pi)$. For any sequence of natural numbers $\{l_k\}_{k=1}^\infty$ it is established that for almost all sequences $\theta=\{\theta_i\}_{i=1}^\infty$ the partial sums $S_{l_k}(x;d\mu_\theta)$ of the Fourier–Stieltjes series of the measure have order $o(\log\log k)$ for almost all $x\in(0;2\pi)$. As proved by Kahane in 1961, the order $o(\log\log k)$ cannot be improved. This result is connected with the well-known problem of Zygmund of finding the exact order of growth of the partial sums of Fourier series almost everywhere.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1994, 78:1, 11–33

Bibliographic databases:

UDC: 517.5
MSC: Primary 60G57; Secondary 42A38
Received: 02.03.1992

Citation: G. A. Karagulian, “On the order of growth $o(\log\log n)$ of the partial sums of Fourier–Stieltjes series of random measures”, Mat. Sb., 184:1 (1993), 15–40; Russian Acad. Sci. Sb. Math., 78:1 (1994), 11–33

Citation in format AMSBIB
\Bibitem{Kar93}
\by G.~A.~Karagulian
\paper On the order of growth $o(\log\log n)$ of the~partial sums of Fourier--Stieltjes series of random measures
\jour Mat. Sb.
\yr 1993
\vol 184
\issue 1
\pages 15--40
\mathnet{http://mi.mathnet.ru/msb954}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1211364}
\zmath{https://zbmath.org/?q=an:0827.42005}
\transl
\jour Russian Acad. Sci. Sb. Math.
\yr 1994
\vol 78
\issue 1
\pages 11--33
\crossref{https://doi.org/10.1070/SM1994v078n01ABEH003456}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1994NR97600002}


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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. G. A. Karagulyan, “On Riemann sums and maximal functions in $\mathbb R^n$”, Sb. Math., 200:4 (2009), 521–548  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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