Some remarks on the unified characterization of reproducing systems

Abstract

The affine systems generated by Ψ ⊂ L2(ℝ n) are the systems AA(Ψ) = {DAj Tk Ψ : j ∈ ℤ ∈ ℤn,} where T k 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(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 A A(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.

Department(s)

Mathematics

Document Type

Article

Keywords

Affine systems, Characterization equations, Tight frames, Wavelets

Publication Date

12-1-2006

Journal Title

Collectanea Mathematica

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