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Generalized multi-resolution analyses and a construction procedure for all wavelet sets in $$\mathbb{R}^n$$. (English) Zbl 0972.42021
Let $$A$$ be a real integer $$n\times n$$ matrix, for which all eigenvalues have absolute value greater than one (i.e., an expansive matrix) and define a unitary operator on $$L^2(R^n)$$ by $$[\delta_Af](x)= |\det A |^{1/2}f(Ax)$$. A set $$W\subseteq R^n$$ is a wavelet set (with respect to $$A$$) if the function $$\psi$$ defined by $$\chi_W=\hat{\psi}$$ is a wavelet (i.e., if $$\{|\det A|^{m/2} \psi(A^m \cdot -k)\}_{m\in Z, k\in Z^n}$$ is an orthonormal basis for $$L^2(R^n)$$). A generalized multiresolution analysis (GMRA) associated to $$A$$ is an increasing family of closed subspaces $$\{V_j\}$$ of $$L^2(R)$$ for which (i) $$V_{j+1}=\delta_A V_j$$ for all $$j$$, (ii) $$\cap V_j = \{0\}$$ and $$\overline {\cup V_j}= L^2(R^n)$$, (iii) $$V_0$$ is invariant under the action of the translation operators $$(\gamma_kf)(x)= f(x+k), k\in Z^n$$. The relationship between GMRA’s and multiwavelets is explored, and as a consequence the authors are able to construct all wavelet sets in $$R^n$$, not only the ones arising from a multiresolution analysis. Several new wavelet sets are constructed.

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