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The existence of subspace wavelet sets. (English) Zbl 1016.42020

Summary: Let \({\mathcal H}\) be a reducing subspace of \(L^2(\mathbb{R}^d)\), that is, a closed subspace of \(L^2(\mathbb{R}^d)\) with the property that \(f(A^m t-\ell)\in{\mathcal H}\) for any \(f\in{\mathcal H}\), \(m\in\mathbb{Z}\) and \(\ell\in\mathbb{Z}^d\), where \(A\) is a \(d\times d\) expansive matrix. It is known that \({\mathcal H}\) is a reducing subspace if and only if there exists a measurable subset \(M\) of \(\mathbb{R}^d\) such that \(A^t M=M\) and \({\mathcal F}({\mathcal H})= L^2(\mathbb{R}^d)\cdot\chi_M\). Under some given conditions of \(M\), it is known that there exist \(A\)-dilation subspace wavelet sets with respect to \({\mathcal H}\). In this paper, we prove that this holds in general.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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