Zeros of the Zak transform on locally compact abelian groups

Abstract

Let G G be a locally compact abelian group. The notion of Zak transform on L 2 ( R d ) L^2(\mathbb {R}^d) extends to L 2 ( G ) L^2(G) . Suppose that G G is compactly generated and its connected component of the identity is non-compact. Generalizing a classical result for L 2 ( R ) L^2(\mathbb {R}) , we then prove that if f ∈ L 2 ( G ) f \in L^2(G) is such that its Zak transform Z f Z f is continuous on G × G ^ G \times \widehat {G} , then Z f Z f has a zero.