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Mat. Sb., 2009, Volume 200, Number 4, Pages 53–82 (Mi msb5328)  

This article is cited in 3 scientific papers (total in 3 papers)

On Riemann sums and maximal functions in $\mathbb R^n$

G. A. Karagulyan

Institute of Mathematics, National Academy of Sciences of Armenia

Abstract: We investigate problems on a.e. convergence of Riemann sums
\begin{equation*} R_nf(x)=\frac1n\sum_{k=0}^{n-1}f(x+\frac kn), \qquad x\in\mathbb T, \end{equation*}
with the use of classical maximal functions in $\mathbb R^n$. A theorem on the equivalence of Riemann and ordinary maximal functions is proved, which allows us to use techniques and results of the theory of differentiation of integrals in $\mathbb R^n$ in these problems. Using this method we prove that for a certain sequence $\{n_k\}$ the Riemann sums $R_{n_k}f(x)$ converge a.e. to $f\in L^p$, $p>1$.
Bibliography: 23 titles.

Keywords: Riemann sums, maximal functions, covering lemmas, sweeping out properties.

DOI: https://doi.org/10.4213/sm5328

Full text: PDF file (699 kB)
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English version:
Sbornik: Mathematics, 2009, 200:4, 521–548

Bibliographic databases:

UDC: 517.518.121
MSC: 42B25, 26A42, 40A30
Received: 13.04.2008

Citation: G. A. Karagulyan, “On Riemann sums and maximal functions in $\mathbb R^n$”, Mat. Sb., 200:4 (2009), 53–82; Sb. Math., 200:4 (2009), 521–548

Citation in format AMSBIB
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\by G.~A.~Karagulyan
\paper On Riemann sums and maximal functions in~$\mathbb R^n$
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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. Preobrazhenskii I.E., “Obobschenie teoremy Iessena o skhodimosti summ Rimana na mnogomernyi sluchai”, Yaroslavskii pedagogich. vestn., 3:4 (2010), 46–52  mathscinet  elib
    2. Karagulyan G.A., “On the sweeping out property for convolution operators of discrete measures”, Proc. Amer. Math. Soc., 139:7 (2011), 2543–2552  crossref  mathscinet  zmath  isi  scopus
    3. Karagulyan G.A., “On Equivalency of Martingales and Related Problems”, J. Contemp. Math. Anal.-Armen. Aca., 48:2 (2013), 51–65  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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