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

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

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

Abstract. We define lacunary Fourier series on a compact connected semisimple Lie group G. If f ∈ L 1 (G) has lacunary Fourier series and f vanishes on a non empty open subset of G, then we prove that f vanishes identically. This result can be viewed as a qualitative uncertainty principle