New proof for Balian-Low theorem of nonlinear Gabor system (Q2443726)

Motivated by the Gabor system \(\{e^{im2\pi t}g(t-n)\}_{m, n\in\mathbb{Z}}\) for \(L^{2}(\mathbb{R})\), the authors consider a new Gabor system \(\{e^{im\theta(2\pi t)}g(t-n)\}_{m, n\in\mathbb{Z}}\), where \(\theta(t)\) satisfies certain assumptions for \(L^{2}(\mathbb{R}, d\theta)\). Then they give a new proof of the Balian-law theorem for the Gabor system \(\{e^{im\theta(2\pi t)}g(t-n)\}_{m, n\in\mathbb{Z}}\). To this purpose they establish relations between the spaces \(L^{2}(\mathbb{R}, d\theta)\) and \(L^{2}(\mathbb{R})\) and they find equivalent conditions for the Gabor system \(\{e^{im\theta(2\pi t)}g(t-n)\}_{m, n\in\mathbb{Z}}\) to be a frame for \(L^{2}(\mathbb{R}, d\theta)\). Also they extend many significant results of the Gabor system for \(L^{2}(\mathbb{R})\) to the Gabor system \(\{e^{im\theta(2\pi t)}g(t-n)\}_{m, n\in\mathbb{Z}}\) for \(L^{2}(\mathbb{R}, d\theta)\).