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Homogeneous approximation property for wavelet frames

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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 S and T 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.

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Correspondence to Wenchang Sun.

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This work was supported partially by the National Natural Science Foundation of China (10571089), The Natural Science Foundation of Tianjin (08JCYBJC09600) and the Program for New Century Excellent Talents in Universities.

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Sun, W. Homogeneous approximation property for wavelet frames. Monatsh Math 159, 289–324 (2010). https://doi.org/10.1007/s00605-008-0055-1

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