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Izv. RAN. Ser. Mat., 1994, Volume 58, Issue 2, Pages 153–166 (Mi izv807)  

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

The Hardy–Littlewood problem for regular and uniformly distributed number sequences

V. A. Oskolkov


Abstract: Let $H$ be the set of functions $f(x)$ defined in $(0, 1)$, $f(0+0)=f(1-0)=+\infty$, monotone in neighborhoods of singular points and such that the improper Riemann integral $\int\limits_0^1f(x) dx$ converges. Let $Q$ be an arbitrary set of sequences $(\{x_i\})_{i=1}^\infty$ uniformly distributed in the interval $[0, 1]$. We find the set of those pairs in $H\times Q$ for which the following equality is valid:
$$ \lim\limits_{n\to\infty}\frac{1}{n}\sum_{i=1}^n f(\{x_i\})=\int\limits_0^1f(x) dx. $$


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English version:
Russian Academy of Sciences. Izvestiya Mathematics, 1995, 44:2, 359–371

Bibliographic databases:

UDC: 511+511.9
MSC: 11J71, 11J83
Received: 17.12.1992

Citation: V. A. Oskolkov, “The Hardy–Littlewood problem for regular and uniformly distributed number sequences”, Izv. RAN. Ser. Mat., 58:2 (1994), 153–166; Russian Acad. Sci. Izv. Math., 44:2 (1995), 359–371

Citation in format AMSBIB
\Bibitem{Osk94}
\by V.~A.~Oskolkov
\paper The Hardy--Littlewood problem for regular and uniformly distributed number sequences
\jour Izv. RAN. Ser. Mat.
\yr 1994
\vol 58
\issue 2
\pages 153--166
\mathnet{http://mi.mathnet.ru/izv807}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1275906}
\zmath{https://zbmath.org/?q=an:0837.11039}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1995IzMat..44..359O}
\transl
\jour Russian Acad. Sci. Izv. Math.
\yr 1995
\vol 44
\issue 2
\pages 359--371
\crossref{https://doi.org/10.1070/IM1995v044n02ABEH001601}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995RB41200008}


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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. Baxa C., “Calculation of Improper Integrals Using Uniformly Distributed Sequences”, Acta Arith., 119:4 (2005), 373–406  crossref  mathscinet  zmath  adsnasa  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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