Un effet de moir\'e sur les espaces sym\'etriques de type non-compact

TL;DR

This paper demonstrates that Helgason waves in noncompact symmetric spaces can be constructed by superimposing elementary spherical functions centered along a horocycle, revealing a moiré effect in the harmonic analysis of such spaces.

Contribution

It introduces a novel method to generate Helgason waves through superposition of spherical functions along a horocycle, highlighting a new geometric phenomenon.

Findings

Helgason waves can be obtained by superimposing spherical functions along a horocycle.

A moiré-like effect occurs in the harmonic analysis of noncompact symmetric spaces.

The approach links geometric configurations to harmonic function construction.

Abstract

We prove that if $X$ is a symmetric space of the noncompact type, just as adding Helgason waves which propagate in all direction yields an elementary spherical function for $X$, a Helgason wave can be produced by adding elementary spherical functions whose centers cluster along a horocycle in $X$.