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Semi-orthogonal frame wavelets and Parseval frame wavelets associated with GMRA. (English) Zbl 1198.42050
Summary: We study semi-orthogonal frame wavelets and Parseval frame wavelets (PFWs) in $$L^{2}(\mathbb{R}^d)$$ with matrix dilations of form $$(Df)(x)=\sqrt 2f(Ax)$$, where $$A$$ is an arbitrary expanding $$d\times d$$ matrix with integer coefficients, such that $$|det A| = 2$$. Firstly, we obtain a necessary and sufficient condition for a frame wavelet to be a semi-orthogonal frame wavelet. Secondly, we present a necessary condition for the semi-orthogonal frame wavelets. When the frame wavelets are the PFWs, we prove that all PFWs associated with generalized multiresolution analysis (GMRA) are equivalent to a closed subspace $$W_{0}$$ for which $$\{T_k\psi :k \in \mathbb{Z}^d\}$$ is a Parseval frame (PF). Finally, by showing the relation between principal shift invariant spaces and their bracket function, we discover a property of the PFWs associated with GMRA by the PFWs’ minimal vector-filter. In each section, we construct concrete examples.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 94A11 Application of orthogonal and other special functions
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