Riesz Wavelets, Tiling and Spectral Sets in LCA Groups

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.