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Towards a characterization of matrices admitting wavelet sets

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Darrin Speegle, SLU

What
  • Analysis Seminar
When Fri, Sep 26, 2014
from 03:10 PM to 04:00 PM
Where RH 237
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Let A be an invertible N × N matrix, and let Γ be a full-rank lattice in RN. An (A, Γ) orthonormal wavelet is a function ψ ∈ L2(RN ) such that

{|A|j/2ψ(Aj x + k) : j ∈ Z, k ∈ Γ}

is an orthonormal basis for L2(RN). There is no characterization of pairs (A,Γ) for which there exists an orthonormal wavelet. In this talk, we consider an ostensibly simpler problem of characterizing the pairs (A, Γ) for which there exists a wavelet set; that is, a set W ⊂ RN such that the indicator function on W is the Fourier transform of an (A,Γ) orthonormal wavelet. Equivalently, we wish to characterize pairs (A,Γ) such that there exists W with {Aj(W):j∈Z} and {W+k:k∈Γ} are measurable tilings of RN. I will reporton the state of the problem and give some modest, new partial results. 

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