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...

• Type: Journal
• Year: 2007
• Where: JAT
• Authors: Christopher Heil, Gitta Kutyniok
• Keywords: Homogeneous Approximation Property, Irregular Wavelet Frames, JAT 2007, Wavelet Frames