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15 July 2002 Wavelets, tiling, and spectral sets
Yang Wang
Duke Math. J. 114(1): 43-57 (15 July 2002). DOI: 10.1215/S0012-7094-02-11413-6

Abstract

We consider a function $\phi\in L\sp 2(\mathbb {R}\sp d)$ such that $\{| \det(D)|\sp {1/2}\phi(Dx-\lambda) : D\in \mathscr {D},\lambda\in \mathscr {T}\}$ forms an orthogonal basis for $L\sp 2(\mathbb {R}\sp d)$, where $\mathscr {D}\subset M\sb d(\mathbb {R})$ and $\mathscr {T}\subset \mathbb {R}\sp d)$. Such a function $\phi$ is called a wavelet with respect to the dilation set $\mathscr {D}$ and translation set $\mathscr {T}$. We study the following question: Under what conditions can a $\mathscr {D}\subset M\sb d(\mathbb {R}$ and a $\mathscr {T}\subset \mathbb {R}\sp d)$ be used as, respectively, the dilation set and the translation set of a wavelet? When restricted to wavelets of the form $\phi=\check{\chi}\Omega$, this question has a surprising tie to spectral sets and their spectra.

Citation

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Yang Wang. "Wavelets, tiling, and spectral sets." Duke Math. J. 114 (1) 43 - 57, 15 July 2002. https://doi.org/10.1215/S0012-7094-02-11413-6

Information

Published: 15 July 2002
First available in Project Euclid: 18 June 2004

zbMATH: 1011.42024
MathSciNet: MR1915035
Digital Object Identifier: 10.1215/S0012-7094-02-11413-6

Subjects:
Primary: ‎42C40
Secondary: 43A45, 52C20, 52C22

Rights: Copyright © 2002 Duke University Press

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Vol.114 • No. 1 • 15 July 2002
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