For any dilation matrix with integral entries $A\in {\mathbb{R}}^{d\times d}$, $d\ge 1$, we construct two families of Parseval wavelet frames in $L^2({\mathbb{R}}^d)$. Both families have compact support and any desired number of vanishing moments. The first family has $|\det A|$ generators. The second family has any desired degree of regularity. For the members of this family, the number of generators depends on the dilation matrix A and the dimension d, but never exceeds $|\det A|+d$. Our construction involves trigonometric polynomials developed by Heller to obtain refinable functions, the Oblique Extension Principle, and a slight generalization of a theorem of Lai and Stöckler.

对于任何具有整数项的膨胀矩阵$\mathrm{一个}\in {\mathbb{R}}^{d\times d}$,$d\ge 1$，我们构造了两个 Parseval 小波框架族${\mathrm{大号}}^2({\mathbb{R}}^d)$. 两个系列都有紧凑的支撑和任意数量的消失时刻。第一个家庭有$|\mathrm{检测}\mathrm{一个}|$发电机。第二个家庭有任何想要的规律性。对于这个家族的成员，生成器的数量取决于扩张矩阵A和维度d，但永远不会超过$|\mathrm{检测}\mathrm{一个}|+d$. 我们的构造涉及由 Heller 开发的三角多项式以获得可细化函数、斜扩展原理以及 Lai 和 Stöckler 定理的轻微推广。