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Frame wavelet sets in \({\mathbb{R}}^{d}\). (English) Zbl 1021.42019
In previous work [“Frame wavelet sets in \(\mathbb{R}\)”, Proc. Am. Math. Soc. 129, 2045-2055 (2001; Zbl 0973.42029)] X. Dai, Y. Diao and Q. Gu constructed wavelet frames \(\{2^{j/2}\psi(2^jx-k)\}_{j,k\in\mathbb{Z}}\) for \(L^2(\mathbb{R})\) generated by functions \(\psi\) having the form \(\hat{\psi}= 1/\sqrt{2\pi}\chi_E\) for some measurable set \(E\subset \mathbb{R}\). Here, the results are extended to \(d\)-dimensions and systems of the form \(\{|\text{det } A |^{n/2} \psi(A^n x -k)\}_{n\in \mathbb{Z}, k\in \mathbb{Z}^d}\), where \(A\) is a real-valued invertible matrix. Equivalent conditions on \(\psi\) generating a tight frame are obtained; and a sufficient, as well as (another) necessary, condition for \(\psi\) generating a frame is given.

MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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