Homogeneous approximation property for wavelet frames

Abstract

The homogeneous approximation property (HAP) of wavelet frames 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 show that every wavelet frame generated with functions satisfying some moderate decay conditions possesses the HAP. Our result improves a recent work of Heil and Kutyniok’s. Moreover, for wavelet frames generated with separable time-scale parameters, i.e., wavelet frames of the form $$\bigcup_{\ell=1}^r\{s^{-d/2}\psi_{\ell}(s^{-1} \cdot - t):\, s\in S_{\ell}, t\in T_{\ell}\},$$ where Sl and Tl are arbitrary sequences of positive numbers and points of \({{\mathbb {R}}^d}\) , respectively, 1≤ l≤ r, we show that the admissibility of wavelet functions is sufficient to guarantee the HAP. Furthermore, we give quantitative results on the approximation error. As consequences of the HAP, we also obtain some density conditions for wavelet frames, which generalize similar results for the case of d = 1.