Geodesic
On Riemann sums and maximal functions in $\mathbb R^n$
Sbornik. Mathematics, Tome 200 (2009) no. 4, pp. 521-548 Cet article a éte moissonné depuis la source Math-Net.Ru

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We investigate problems on a.e. convergence of Riemann sums \begin{equation*} R_nf(x)=\frac1n\sum_{k=0}^{n-1}f\biggl(x+\frac kn\biggr), \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. 
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     title = {On {Riemann} sums and maximal functions in~$\mathbb R^n$},
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G. A. Karagulyan. On Riemann sums and maximal functions in $\mathbb R^n$. Sbornik. Mathematics, Tome 200 (2009) no. 4, pp. 521-548. http://geodesic.mathdoc.fr/item/SM_2009_200_4_a2/

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