The Balian-Low theorem for the windowed quaternionic Fourier transform
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.
scientific article; zbMATH DE number 6111295
scientific article