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Riesz Wavelets, Tiling and Spectral Sets in LCA Groups

  • Azita Mayeli
Article
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Abstract

Let G be a locally compact abelian (LCA) group equipped with a Haar measure. A collection of measurable subsets \(\{\Omega _i\}_{i=1}^m\) in \(\hat{G}\) is called a Riesz wavelet collection if there are countable subsets \(A\subset \mathcal {A}ut(G)\) and \(\Lambda _i\subset G\) such that for \(\hat{\psi }_i:= 1_{\Omega _i}\), the family
$$\begin{aligned} \mathcal {W}:=\cup _{i=1}^m\{\Delta (a)^{1/2} \psi _i(a(x)-\lambda ): \ \lambda \in \Lambda _i, a\in A\} \end{aligned}$$
is a Riesz basis for \(L^2(G)\). In this paper we show that if \(\Omega _i\)’s are Riesz spectral sets and \(\Omega =\cup _i \Omega _i\) is a multiplicative tiling set for \(\hat{G}\), then \(\mathcal {W}\) is a Riesz basis for \(L^2(G)\). The converse also holds if the unit element \(e\in G\) belongs to all \(\Lambda _i\). As a result, if \(\Omega _i\)’s multi-tile \(\hat{G}\) additively by lattice and \(\Omega \) is a multiplicative tiling, then \(\mathcal {W}\) is a Riesz basis for \(L^2(G)\). When \(m=1\), we show that the multiplicative tiling property of \(\Omega \) is equivalent to the Riesz spectral property of \(\hat{\alpha }(\Omega )\), \(\alpha \in A\), provided that \(\mathcal {W}\) is a Riesz basis.

Keywords

Riesz bases Wavelets Spectral and tiling 

Mathematics Subject Classification

Primary 43A25 43A46 42C40 Secondary 52C22 

Notes

Acknowledgements

Support for this project was partially provided by PSC-CUNY by PSC-CUNY Award B \(\sharp \) 69625-00 47, jointly funded by The Professional Staff Congress and The City University of New York.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.City University of New YorkNew YorkUSA

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