Beurling's Theorem and Characterization of Heat Kernel for Riemannian Symmetric Spaces of Noncompact Type We prove Beurling's theorem for rank $1$ Riemannian symmetric spaces and relate its consequences with the characterization of the heat kernel of the symmetric space. Keywords:Beurling's Theorem, Riemannian symmetric spaces, uncertainty principleCategories:22E30, 43A85