On Riemann sums and maximal functions in $\ZR^n$

TL;DR

This paper explores the almost everywhere convergence of Riemann sums in multiple dimensions, establishing a connection with maximal functions and proving convergence for functions in L^p spaces with p>1.

Contribution

It introduces a novel link between Riemann sums and maximal functions, enabling the use of differentiation theory techniques in higher dimensions.

Findings

Almost everywhere convergence of Riemann sums for p>1

Connection established between Riemann sums and maximal functions

Convergence results extend to infinite-dimensional sequences

Abstract

In this paper we investigate problems on almost everywhere convergence of subsequences of Riemann sums \md0 R_nf(x)=\frac{1}{n}\sum_{k=0}^{n-1}f\bigg(x+\frac{k}{n}\bigg),\quad x\in \ZT. \emd We establish a relevant connection between Riemann and ordinary maximal functions, which allows to use techniques and results of the theory of differentiations of integrals in $\ZR^n$ in mentioned problems. In particular, we prove that for a definite sequence of infinite dimension $n_k$ Riemann sums $R_{n_k}f(x)$ converge almost everywhere for any $f\in L^p$ with $p>1$.