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Semi-orthogonal parseval frame wavelets and generalized multiresolution analyses. (English) Zbl 1106.42026
Summary: We study Parseval frame wavelets in $$L^2(\mathbb{R}^d)$$ with matrix dilations of the form $$(Df)(x)=\sqrt 2f(Ax)$$, where $$A$$ is an arbitrary expanding $$n\times n$$-matrix with integer coefficients, such that $$|\det A|=2$$. In our study we use generalized multiresolution analyses (GMRA) $$(V_j)$$ in $$L^2(\mathbb{R}^d)$$ with dilations $$D$$. We describe, in terms of the underlying multiresolution structure, all GMRA Parseval frame wavelets and, a posteriori, all semi-orthogonal Parseval frame wavelets in $$L^2(\mathbb{R}^d)$$. As an application, we include an explicit construction of an orthonormal wavelet on the real line whose dimension function is essentially unbounded.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
##### Keywords:
parseval frame; wavelet; multiresolution analysis
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##### References:
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