Homogeneous Approximation Property for Wavelet Frames

The homogeneous approximation property (HAP) is useful in practice since it means that the number of building blocks involved in a reconstruction of f up to some error is essentially invariant under time-scale shifts. In this paper, we prove the HAP for a large class of wavelet frames which are generated with arbitrary time-scale shifts and wavelet functions satisfying moderate decaying condition. Our result improves a recent work by Heil and Kutyniok. Moreover, for wavelet frames generated with irregular affine lattices, i.e., wavelet frames of the form r ∪ l=1 {sψl(s · −t) : s ∈ Sl, t ∈ Tl}, where Sl and Tl are arbitrary sequences of positive numbers and points in Rd, respectively, 1 ≤ l ≤ r, we show that the admissibility of wavelet functions is enough to guarantee the homogeneous approximation property. Furthermore, we give quantitative results on the approximation error. As consequences of the HAP, we also get some density conditions for wavelet frames, which generalizes similar results for the case of d = 1.