an:06825637 Zbl 1391.42036 Hardin, Douglas P.; Northington, Michael C.; Powell, Alexander M. A sharp Balian-Low uncertainty principle for shift-invariant spaces EN Appl. Comput. Harmon. Anal. 44, No. 2, 294-311 (2018). 00373160 2018
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42C15 Balian-Low theorem; shift-invariant spaces; uncertainty principle The authors prove a sharp version of the Balian-Low theorem for the generators of finitely generated shift-invariant spaces. For singly generated shift-invariant spaces their main results take the following form: 1.(Corollary 1.4) Fix lattices $$\Lambda,\Gamma\subset\mathbb{R}^d$$ with $$\Lambda\subset\Gamma$$ and with the index of the lattice $$\Lambda$$ in $$\Gamma$$ greater than 1 ($$[\Gamma\, : \,\Lambda]>1).$$ Suppose $$f\in L^2(\mathbb{R}^d),$$ $$\| f\|_2\neq 0,$$ and $$\mathcal{T}^{\Lambda}(f)=\{f(\cdot -\lambda)\, :\,\lambda\in\Lambda\}$$ forms a frame for the closed linear span $$V^{\Lambda}(f)$$ of $$\mathcal{T}^{\Lambda}(f)$$ in $$L^2(\mathbb{R}^d).$$ If $$V^{\Lambda}(f)$$ is $$\Gamma -$$ invariant, then $$\int_{L^2(\mathbb{R}^d)}|x||f(x)|^2dx=\infty.$$ 2.(Corollary 1.6) Fix a lattice $$\Lambda\subset\mathbb{R}^d.$$ Suppose $$f\in L^2(\mathbb{R}^d)$$ with $$\| f\|_2\neq 0.$$ If $$\mathcal{T}^{\Lambda}(f)$$ is a frame for $$V^{\Lambda}(f) ,$$ but is not a Riesz basis for $$V^{\Lambda}(f),$$ then $$\int_{L^2(\mathbb{R}^d)}|x||f(x)|^2dx=\infty.$$ The main results provide an absolutely sharp improvement of the best previously existing results in the literature in this direction. Alexei Lukashov (Saratov)