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Parseval frame wavelets with \(E_{n}^{(2)}\)-dilations. (English) Zbl 1090.42020
Let \(A\) be an expanding \(n\times n\) matrix with integer entries and \(| \det(A)| =2\). Given a function \(\psi\in L^2(\mathbb R^n)\), consider the associated wavelet system \(\Psi=\{2^{j/2}\psi(A^j\cdot-k)\}_{j\in Z, k\in Z^n}.\) It is shown that each multiresolution analysis generated by the matrix \(A\) admits Parseval (multi)-wavelet frames (i.e., tight frames), generated by either one or two functions. The minimal number of generators is determined. All Parseval frames associated with the multiresolution analysis generated by the matrix \(A\) are characterized.

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