The Balian-Low theorem for the windowed quaternionic Fourier transform
type: Item
instance of: scholarly article
title: The Balian-Low theorem for the windowed quaternionic Fourier transform
zbMATH Open document ID: 1256.42044
author: Yingxiong Fu
author: Paula Cerejeiras
author: Uwe Kähler
published in: Advances in Applied Clifford Algebras
publication date: 2012-11-29T00:00:00Z
review text: Following \textit{M. Shapiro} and \textit{L. M. Tovar} in [Contemp. Math. 212, 229-244; (1998; Zbl 0907.30047)] the authors define the right multiplicative operator \(\mathcal C[\lambda, \cdot]\). The kernel of the windowed quaternionic Fourier transform (WQFT) is then expressed as \(g_{\omega,\mathbf b}:= e^{2\pi\mathbf j\omega_2x_2}g(\mathbf x-\mathbf b)\mathcal C[e^{2\pi \mathbf i\omega_1x_1},\cdot]\), \(\omega,\mathbf b, \mathbf x\in{\mathbb R}^2\), and the definition of the WQFT is established for the given window function \(g\in L^2(\mathbb{R}^2;\mathbb{R})\). Properties of the WQFT are derived such as shift, modulation, reconstruction formula, orthogonality relation, and Heisenberg uncertainty principle. Considering the discrete version of the kernel of the inverse WQFT, they establish a new Gabor system \(\overline{g_{m,n}}:= \mathcal C[\cdot, e^{2\pi\mathbf i m_1x_1}]g(\mathbf x-\mathbf n)e^{2\pi\mathbf j m_2x_2}\), \(\mathbf x\in \mathbb{R}^2,\mathbf m, \mathbf n\in\mathbb{Z}^2\), which induces an orthonormal basis with respect to the scalar inner product. The corresponding Balian--Low theorem is also proved, which can be regarded as a strong version of the Heisenberg uncertainty principle for Gabor systems. Moreover, it is pointed out that this theorem also holds true for the right-sided WQFT for a real-valued window function.
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 42A38
Mathematics Subject Classification ID: 15A66
zbMATH DE Number: 6111295
zbMATH Keywords: Gabor system
zbMATH Keywords: Balian-Low theorem
zbMATH Keywords: quaternionic Fourier transform
zbMATH Keywords: windowed quaternionic Fourier transform
MaRDI profile type: Publication
full work available at URL: s00006-012-0324-x
OpenAlex ID: W1995933071
Wikidata QID: Q58038897
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DOI: 10.1007/S00006-012-0324-X
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description: scientific article; zbMATH DE number 6111295