# Some remarks on the unified characterization of reproducing systems.

Collectanea Mathematica (2006)

- Volume: 57, Issue: 3, page 295-307
- ISSN: 0010-0757

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topGuo, Kanghui, and Labate, Demetrio. "Some remarks on the unified characterization of reproducing systems.." Collectanea Mathematica 57.3 (2006): 295-307. <http://eudml.org/doc/41774>.

@article{Guo2006,

abstract = {The affine systems generated by Ψ ⊂ L2(Rn) are the systemsAA(Ψ) = \{DjA Tk Ψ : j ∈ Z, k ∈ Zn\},where Tk are the translations, and DA 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 L2(Rn). 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 AA(Ψ) 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.},

author = {Guo, Kanghui, Labate, Demetrio},

journal = {Collectanea Mathematica},

keywords = {Análisis de Fourier; Matrices de dilatación; Ondículas},

language = {eng},

number = {3},

pages = {295-307},

title = {Some remarks on the unified characterization of reproducing systems.},

url = {http://eudml.org/doc/41774},

volume = {57},

year = {2006},

}

TY - JOUR

AU - Guo, Kanghui

AU - Labate, Demetrio

TI - Some remarks on the unified characterization of reproducing systems.

JO - Collectanea Mathematica

PY - 2006

VL - 57

IS - 3

SP - 295

EP - 307

AB - The affine systems generated by Ψ ⊂ L2(Rn) are the systemsAA(Ψ) = {DjA Tk Ψ : j ∈ Z, k ∈ Zn},where Tk are the translations, and DA 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 L2(Rn). 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 AA(Ψ) 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.

LA - eng

KW - Análisis de Fourier; Matrices de dilatación; Ondículas

UR - http://eudml.org/doc/41774

ER -

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