Homogeneous approximation property for wavelet frames

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

$bigcup_{ell=1}^r{s^{-d/2}psi_{ell}(s^{-1} cdot - t):, sin S_{ell}, tin T_{ell}},$bigcup_{ell=1}^r{s^{-d/2}psi_{ell}(s^{-1} cdot - t):, sin S_{ell}, tin T_{ell}},
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