376 research outputs found

    Wavelets for non-expanding dilations and the lattice counting estimate

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    We show that problems of existence and characterization of wavelets for non-expanding dilations are intimately connected with the geometry of numbers; more specifically, with a bound on the number of lattice points in balls dilated by the powers of a dilation matrix AGL(n,R)A \in \mathrm{GL}(n,\mathbb{R}). This connection is not visible for the well-studied class of expanding dilations since the desired lattice counting estimate holds automatically. We show that the lattice counting estimate holds for all dilations AA with detA1\left|\det{A}\right|\ne 1 and for almost every lattice Γ\Gamma with respect to the invariant probability measure on the set of lattices. As a consequence, we deduce the existence of minimally supported frequency (MSF) wavelets associated with such dilations for almost every choice of a lattice. Likewise, we show that MSF wavelets exist for all lattices and and almost every choice of a dilation AA with respect to the Haar measure on GL(n,R)\mathrm{GL}(n,\mathbb{R})

    Oversampling of wavelet frames for real dilations

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    We generalize the Second Oversampling Theorem for wavelet frames and dual wavelet frames from the setting of integer dilations to real dilations. We also study the relationship between dilation matrix oversampling of semi-orthogonal Parseval wavelet frames and the additional shift invariance gain of the core subspace.Comment: Journal of London Mathematical Society, published online March 13, 2012 (to appear in print
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