Framelets derived from refinable (vector) functions via the popular oblique extension principle (OEP) are of interest in both theory and applications. Though OEP can increase vanishing moments of framelet generators to improve sparsity, it has a serious shortcoming for scalar framelets: the associated discrete framelet transform is often not compact and deconvolution is unavoidable. On the other hand, in sharp contrast to the extensively studied scalar framelets, OEP-based multiframelets are far from well understood. In this paper, we prove that from any compactly supported refinable vector function having at least two entries, one can always construct through OEP a compactly supported quasi-tight multiframelet such that all framelet generators have the highest possible order of vanishing moments, and its underlying discrete framelet transform is compact and balanced. The key ingredient of our proof is a newly developed normal form of matrix-valued filters, which greatly facilitates the study of multiframelets.

通过流行的倾斜扩展原理（OEP）从可精化（矢量）函数派生的小框架在理论和应用中都引起人们的兴趣。尽管OEP可以增加小框架生成器的消失力矩以改善稀疏性，但是它对于标量小框架具有严重的缺点：关联的离散小框架变换通常不紧凑，并且反卷积是不可避免的。另一方面，与广泛研究的标量小框架形成鲜明对比的是，基于OEP的多小框架远未得到很好的理解。在本文中，我们证明，从任何至少有两个入口的紧凑支持的可精炼向量函数中，一个人总是可以通过OEP构造一个紧凑支持的准紧复框架，从而使所有框架生成器都具有最高的消失矩阶，并且其底层离散小框架变换是紧凑且平衡的。我们证明的关键要素是矩阵值滤波器的一种新开发的标准形式，它极大地促进了多框架的研究。