Some remarks on the unified characterization of reproducing systems
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Autores: Kanghui Guo, Demetrio Labate
- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 57, Fasc. 3, 2006, págs. 295-307
- Idioma: español
- Enlaces
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Resumen
The affine systems generated by $\Psi \subset L^2(\mathbb{R}^n)$ are the systems $\mathcal{A}_A(\Psi) = \{D^j_A T_k\Psi : j \in \mathbb{Z},k \in \mathbb{Z}^n$ where $T_k$ are the translations, and $D_A$ the dilations with respect to an invertible matrix A. As shown in [5], there is a simple characterization for those affine systems that are a Parseval frame for $L^2(\mathbb{R}^n)$. In this paper, we correct an error in the proof of the characterization result from [5], by redefining the class of not-necessarily expanding dilation matrices for which this characterization result holds. In addition, we examine the connection between the eigenvalues of the dilation matrix A and the characterization equations of the affine system $\mathcal{A}_A(\Psi)$ that are Parseval frames. Our observations go in the same directions as other recent results in the literature that show that, when A is not expanding, the information about the eigenvalues alone is not sufficient to characterize or to determine existence of those affine systems that are Parseval frames.
