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.

中文翻译：

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关于具有$$ E_d ^ {（2）}（\ mathbb {Z}）$$ d（2）（Z）扩张的消失矩的对称紧支撑小波

设A为\（{{\ mathbb {R}}} ^ d \）上的展开线性映射，其中保留整数晶格并带有\（| \ det A | = 2 \）。我们证明，如果存在与A相关联的自仿射图块集，则将存在一个紧密支持的小波，该小波具有任意数量的消失矩和一些对称性。我们把重点放在结构关联到线性地图小波甲上\（{{\ mathbb {R}}} ^ 2 \）和所述梅花形扩张上\（{{\ mathbb {R}}} ^ 3 \）因为我们可以消除自仿射图块集存在的假设。我们的构造基于Han的一维低通滤波器，并采用二进位扩张和多分辨率理论。最后，对于某些特定的膨胀矩阵，我们认识到，一维Daubechies低通滤波器可以适用于获得具有任意所需规则度和任意固定消失力矩的紧凑支持小波。