Lacunary Fourier series and a qualitative uncertainty principle for compact Lie groups

TL;DR

This paper establishes a qualitative uncertainty principle for functions on compact Lie groups, showing that lacunary Fourier series functions that vanish on an open set must be identically zero.

Contribution

It introduces the concept of lacunary Fourier series on compact Lie groups and proves a unique continuation property for such functions.

Findings

Lacunary Fourier series functions vanish on open sets only if they are zero everywhere.

The result extends uncertainty principles to the setting of compact Lie groups.

Provides a new perspective on harmonic analysis in non-abelian groups.

Abstract

We define lacunary Fourier series on a compact connected semisimple Lie group $G$. If $f \in L^1(G)$ has lacunary Fourier series, and vanishes on a non empty open set, then we prove that $f$ vanishes identically. This may be viewed as a qualitative uncertainty principle.