In this paper, we study scalar multivariate non-stationary subdivision schemes with integer dilation matrix M and present a unifying, general approach for checking their convergence and for determining their Hölder regularity (latter in the case $$M = mI, m \ge 2$$M=mI,m≥2). The combination of the concepts of asymptotic similarity and approximate sum rules allows us to link stationary and non-stationary settings and to employ recent advances in methods for exact computation of the joint spectral radius. As an application, we prove a recent conjecture by Dyn et al. on the Hölder regularity of the generalized Daubechies wavelets. We illustrate our results with several examples.

中文翻译：

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非平稳细分的规律：矩阵方法

在本文中，我们研究了具有整数膨胀矩阵 M 的标量多元非平稳细分方案，并提出了一种统一的通用方法来检查它们的收敛性和确定它们的 Hölder 正则性（后者在这种情况下 $$M = mI, m \ge 2 $$M=m1，m≥2）。渐近相似性概念和近似求和规则的结合使我们能够将平稳和非平稳设置联系起来，并利用方法的最新进展来精确计算联合谱半径。作为一个应用，我们证明了 Dyn 等人最近的一个猜想。关于广义 Daubechies 小波的 Hölder 正则性。我们用几个例子来说明我们的结果。