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On dual wavelet tight frames. (English) Zbl 0880.42017
Summary: A characterization of multivariate dual wavelet tight frames for any general dilation matrix is presented in this paper. As an application, Lawton’s result on wavelet tight frames in \(L^2(\mathbb{R})\) is generalized to the \(n\)-dimensional case. Two ways of constructing certain dual wavelet tight frames in \(L^2(\mathbb{R}^n)\) are suggested. Finally, examples of smooth wavelet tight frames in \(L^2(\mathbb{R})\) and \(H^2(\mathbb{R})\) are provided. In particular, an example is given to demonstrate that there is a function \(\psi\) whose Fourier transform is positive, compactly supported, and infinitely differentiable which generates a non-MRA wavelet tight frame in \(H^2(\mathbb{R})\).

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