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On Symmetric Compactly Supported Wavelets with Vanishing Moments Associated to \(E_d^{(2)}(\mathbb {Z})\) Dilations

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Abstract

Let A be an expansive linear map on \({{\mathbb {R}}}^d\) preserving the integer lattice and with \(| \det A|=2\). We prove that if there exists a self-affine tile set associated to A, there exists a compactly supported wavelet with any desired number of vanishing moments and some symmetry. We put emphasis on construction of wavelets associated to a linear map A on \({{\mathbb {R}}}^2\) and to the Quincunx dilation on \({{\mathbb {R}}}^3\) because we can remove the hypothesis of the existence of the self-affine tile set. Our construction is based on low pass filters by Han in dimension one with the dyadic dilation and multiresolution theory. Finally, for some particular dilation matrices, we realize that unidimensional Daubechies low pass filers can be adapted to obtain compactly supported wavelets with any desired degree of regularity and any fix number of vanishing moments.

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  1. Departamento de Matemáticas, Universidad de Alicante, 03080, Alicante, Spain

    M. L. Arenas & Angel San Antolín

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  1. M. L. Arenas
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  2. Angel San Antolín
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Correspondence to Angel San Antolín.

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Communicated by Stephan Dahlke.

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Arenas, M.L., San Antolín, A. On Symmetric Compactly Supported Wavelets with Vanishing Moments Associated to \(E_d^{(2)}(\mathbb {Z})\) Dilations. J Fourier Anal Appl 26, 72 (2020). https://doi.org/10.1007/s00041-020-09782-2

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