The Homogeneous Approximation Property for wavelet frames

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The Homogeneous Approximation Property for wavelet frames
An irregular wavelet frame has the form W(ψ, Λ) = {a−1/2 ψ(x a − b)}(a,b)∈Λ, where ψ ∈ L2 (R) and Λ is an arbitrary sequence of points in the affine group A = R+ × R. Such irregular wavelet frames are poorly understood, yet they arise naturally, e.g., from sampling theory or the inevitability of perturbations. This paper proves that irregular wavelet frames satisfy a Homogeneous Approximation Property, which essentially states that the rate of approximation of a wavelet frame expansion of a function f is invariant under time-scale shifts of f, even though Λ is not required to have any structure—it is only required that the wavelet ψ have a modest amount of time-scale concentration. It is shown that the Homogeneous Approximation Property has several implications on the geometry of Λ, and in particular a relationship between the affine Beurling density of the frame and the affine Beurling density of any other Riesz basis of wavelets is derived. This further yields ne...
Christopher Heil, Gitta Kutyniok
Added 15 Dec 2010
Updated 15 Dec 2010
Type Journal
Year 2007
Where JAT
Authors Christopher Heil, Gitta Kutyniok
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