Riesz means on homogeneous trees

TL;DR

This paper proves that Riesz means on homogeneous trees converge almost everywhere for functions in L^p spaces when the real part of the parameter z is positive, extending harmonic analysis results to tree structures.

Contribution

It establishes almost everywhere convergence of Riesz means on homogeneous trees for a range of L^p functions, generalizing classical Euclidean results to tree graphs.

Findings

Almost everywhere convergence of Riesz means for p in [1,2]

Convergence holds when Re z > 0

Extends harmonic analysis to homogeneous trees

Abstract

Let $\mathbb{T}$ be a homogeneous tree. 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>0 $.