376 research outputs found
Wavelets for non-expanding dilations and the lattice counting estimate
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 .
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 with
and for almost every lattice 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 with respect to the Haar measure on
Oversampling of wavelet frames for real dilations
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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