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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
##### Keywords:
wavelet frames; tight frame
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##### References:
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