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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.

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
Full Text: DOI
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