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A sharp Balian-Low uncertainty principle for shift-invariant spaces. (English) Zbl 1391.42036
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.

MSC:
42C15 General harmonic expansions, frames
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