Spherical Averages on Regular and Semiregular Graphs

TL;DR

This paper extends classical results about spherical averages from hyperbolic surfaces to functions on regular and semiregular graphs, analyzing convergence and rates over various geometric sets.

Contribution

It establishes convergence results for averages on graphs, including rates, and explores more general averaging sets like arcs and horocycles.

Findings

Averages over spheres on graphs converge to the mean value.

Convergence rates are explicitly characterized.

Results apply to various geometric averaging sets.

Abstract

In 1966, P. Guenther proved the following result: Given a continuous function f on a compact surface M of constant curvature -1 and its periodic lift g to the universal covering, the hyperbolic plane, then the averages of the lift g over increasing spheres converge to the average of the function f over the surface M. In this article, we prove similar results for functions on the vertices and edges of regular and semiregular graphs, with special emphasis on the convergence rate. We also consider averages over more general sets like arcs, tubes and horocycles.