New Proof for Balian-Low Theorem of Nonlinear Gabor System

Abstract

The main purpose of this paper is to give a new proof of the Balian-Low theorem for Gabor system {eimθ(2πt)g(t−n), m,n∈ℤ}, which is proposed by Fu et al. and associated with nonlinear Fourier atoms. To this end, we establish the relationships between spaces L2(ℝ,dθ) and L2(ℝ). We also introduce the concept of frame associated with nonlinear Fourier atoms for L2(ℝ,dθ) and obtain many subsidiary results for this kind of (Gabor) frames.