An endpoint Balian-Low theorem for Schauder bases

Abstract

We prove two new results on the validity of the Balian-Low theorem in the setting of Schauder bases. Our first main result proves an endpoint Balian-Low theorem for Gabor systems that form Schauder bases for L2(R): if g∈L2(R) is compactly supported and ∫|ξ|2|gˆ(ξ)|2dξ<∞ then the Gabor system G(g,1,1) cannot be a Schauder basis of so-called type Λ. Our second main result proves that the classical Balian-Low theorem for orthonormal bases and Riesz bases fails in the setting of Schauder bases. In particular, given ϵ>0, there exists g∈L2(R) such that G(g,1,1) is a Schauder basis for L2(R) and such that ∫|t|3−ϵ|g(t)|2dt<∞ and ∫|ξ|3−ϵ|gˆ(ξ)|2dξ<∞.