三角多项式的分解及其在多变量细分格式中的应用
Decompositions of Trigonometric Polynomials with Applications to Multivariate Subdivision Schemes
AI总结：
本文基于多相表示，针对满足Strang-Fix条件的多元三角多项式提出一种构造性分解方法，并应用于任意膨胀矩阵下多变量细分格式的收敛性与光滑性分析，给出了充分条件。
摘要：
我们研究满足一组接近已知Strang-Fix条件的多元三角多项式。基于这些多项式相对于一般膨胀矩阵的多相表示，我们开发了一种简单的构造性方法，用于这类多项式的特殊类型分解。这些分解对于分析与一般膨胀矩阵相关的多变量细分格式的收敛性和光滑性具有重要意义。我们通过验证此处证明的多变量标量细分格式收敛性和光滑性的充分条件，应用了这些分解。对于收敛性分析，我们的充分条件适用于任意膨胀矩阵，而先前已知的充要条件仅适用于具有自相似铺砌的膨胀矩阵。对于光滑性分析，我们陈述并证明了关于多变量矩阵细分格式的两个定理，这些定理导出了与各向同性膨胀矩阵相关的标量多变量细分格式的C^1极限的充分条件。尽管文献中陈述了类似结果，但我们在本文中给出了这些结果的详细证明，这些证明在其他地方未能找到。
英文摘要：
We study multivariate trigonometric polynomials, satisfying a set of constraints close to the known Strung-Fix conditions. Based on the polyphase representation of these polynomials relative to a general dilation matrix, we develop a simple constructive method for a special type of decomposition of such polynomials. These decompositions are of interest to the analysis of convergence and smoothness of multivariate subdivision schemes associated with general dilation matrices. We apply these decompositions, by verifying sufficient conditions for the convergence and smoothness of multivariate scalar subdivision schemes, proved here. For the convergence analysis our sufficient conditions apply to arbitrary dilation matrices, while the previously known necessary and sufficient conditions are relevant only in case of dilation matrices with a self similar tiling. For the analysis of smoothness, we state and prove two theorems on multivariate matrix subdivision schemes, which lead to sufficient conditions for C^1 limits of scalar multivariate subdivision schemes associated with isotropic dilation matrices. Although similar results are stated in the literature, we give here detailed proofs of the results, which we could not find elsewhere.
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