---
abstract: |
    Let $X$ be a non-compact symmetric space of dimension $n$. We prove that if $f\in L^{p}(X)$, $1\leq p\leq 2$, then the Riesz means $S_{R}^{z}\left(
    f\right) $ converge to $f$ almost everywhere as $R\rightarrow \infty $, whenever $\operatorname{Re}z>\left( n-\frac{1}{2}\right) \left( \frac{2}{p}-1\right) $.
author:
- 'A. Fotiadis, M. Marias, and E. Papageorgiou'
title:
- Riesz means on symmetric spaces
- Riesz means on symmetric spaces
---

Introduction and statement of the results
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The Riesz means $S_{R}^{z}\left( f\right) $, $R>0$, $\operatorname{Re}z\geq 0$, of a function $f$ defined on the cube $\left[ 0,1\right] ^{n}$, are given by $$S_{R}^{z}\left( f\right) \left( \theta \right) =\sum_{\Vert k\Vert <R}\left[
\left( 1-\frac{\left\Vert k\right\Vert ^{2}}{R^{2}}\right) \right] _{+}^{z}%
\hat{f}\left( k\right) e^{2\pi i\left( k,\theta \right) }\text{, \ }k\in
\mathbb{Z}^{n}\text{, }\theta \in \left[ 0,1\right] ^{n},$$where $\hat{f}$ is the Fourier transform of $f$. If $z=0$, then they are just the partial sums $S_{R}\left( f\right) \left( \theta \right)
=\sum_{\left\Vert k\right\Vert <R}\hat{f}\left( k\right) e^{2\pi i\left(
k,\theta \right) }$ of the multiple Fourier series of $f$. Riesz means are used in order to understand the strange behavior of $S_{R}\left( f\right) $ as $R\longrightarrow \infty $. In fact, if $f\in L^{p}\left( \left[ 0,1%
\right] \right) $, $p\in \left( 1,\infty \right) $, then M. Riesz 