Construction of multivariate tight framelets is known to be a challenging problem because it is linked to the difficult problem on sum of squares of multivariate polynomials in real algebraic geometry. Multivariate dual framelets with vanishing moments generalize tight framelets and are not easy to be constructed either, since their construction is related to syzygy modules and factorization of multivariate polynomials. On the other hand, compactly supported multivariate framelets with directionality or high vanishing moments are of interest and importance in both theory and applications. In this paper we introduce the notion of a quasi-tight framelet, which is a dual framelet, but behaves almost like a tight framelet. Let $\varphi \in L_2({\mathbb{R}}^d)$ be an arbitrary compactly supported real-valued $\mathsf{M}$-refinable function with a general dilation matrix $\mathsf{M}$ and $\hat{\varphi }(0)=1$ such that its underlying real-valued low-pass filter satisfies the basic sum rule. We first constructively prove by a step-by-step algorithm that we can always easily derive from the arbitrary $\mathsf{M}$-refinable function ϕ a directional compactly supported real-valued quasi-tight $\mathsf{M}$-framelet in $L_2({\mathbb{R}}^d)$ associated with a directional quasi-tight $\mathsf{M}$-framelet filter bank, each of whose high-pass filters has one vanishing moment and only two nonzero coefficients. If in addition all the coefficients of its low-pass filter are nonnegative, then such a quasi-tight $\mathsf{M}$-framelet becomes a directional tight $\mathsf{M}$-framelet in $L_2({\mathbb{R}}^d)$. Furthermore, we show by a constructive algorithm that we can always derive from the arbitrary $\mathsf{M}$-refinable function ϕ a compactly supported quasi-tight $\mathsf{M}$-framelet in $L_2({\mathbb{R}}^d)$ with the highest possible order of vanishing moments. We shall also present a result on quasi-tight framelets whose associated high-pass filters are purely differencing filters with the highest order of vanishing moments. Several examples will be provided to illustrate our main theoretical results and algorithms in this paper.

已知构造多元紧密小框架是一个具有挑战性的问题，因为它与实际代数几何中的多元多项式平方和的难题相联系。具有消失矩的多元对偶小框架泛化了紧的框架，也不容易构造，因为它们的构造与syzygy模块和多元多项式的分解有关。另一方面，具有方向性或高消失力矩的紧密支撑的多元框架在理论和应用中都受到关注和重视。在本文中，我们介绍了准紧框架的概念，它是对偶框架，但其行为几乎类似于紧框架。让$\varphi \in {\mathrm{大号}}_2（{\mathbb{[R}}^d）$ 是任意紧凑支持的实值 $\mathsf{中号}$具有一般扩张矩阵的可精炼函数 $\mathsf{中号}$ 和 $\hat{\varphi }（0）=\mathrm{1个}$因此其基础实值低通滤波器满足基本求和规则。我们首先通过逐步算法来建设性地证明，我们总是可以轻松地从任意$\mathsf{中号}$完善的功能ϕ方向性紧致支持的实值准紧$\mathsf{中号}$-framelet ${\mathrm{大号}}_2（{\mathbb{[R}}^d）$ 与定向准紧相关 $\mathsf{中号}$-framelet滤波​​器组，每个高通滤波器具有一个消失的矩，并且只有两个非零系数。另外，如果其低通滤波器的所有系数均为非负值，则这种准紧$\mathsf{中号}$-framelet变成方向性紧 $\mathsf{中号}$-framelet ${\mathrm{大号}}_2（{\mathbb{[R}}^d）$。此外，我们通过构造性算法表明，我们始终可以从任意$\mathsf{中号}$-完善的功能ϕ紧密支撑的准密封$\mathsf{中号}$-framelet ${\mathrm{大号}}_2（{\mathbb{[R}}^d）$消失的可能性最高的顺序。我们还将在准紧小框架上给出一个结果，该准紧小框架的相关高通滤波器是具有最高阶消失力矩的纯差分滤波器。本文将提供几个例子来说明我们的主要理论结果和算法。