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- W2963935146 abstract "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 |$A in mathrm{GL}(n,mathbb{R})$|. 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 |$A$| with |${leftlvert{det{A}}rightrvert}ne 1$| and for almost every lattice |$Gamma$| 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 almost every choice of a dilation |$A$| with respect to the Haar measure on |$mathrm{GL}(n,mathbb{R})$|." @default.
- W2963935146 created "2019-07-30" @default.
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- W2963935146 date "2016-10-31" @default.
- W2963935146 modified "2023-09-27" @default.
- W2963935146 title "Wavelets for Non-expanding Dilations and the Lattice Counting Estimate" @default.
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- W2963935146 doi "https://doi.org/10.1093/imrn/rnw222" @default.
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