The regularity of refinable functions has been analysed in an extensive literature and is well-understood in two cases: 1) univariate 2) multivariate with an isotropic dilation matrix. The general (non-isotropic) case offered a great resistance. It was not before 2019 that the non-isotropic case was done by developing the matrix method. In this paper we make the next step and extend the Littlewood-Paley type method, which is very efficient in the aforementioned special cases, to general equations with arbitrary dilation matrices. This gives formulas for the higher order regularity in W2k(Rn) by means of the Perron eigenvalue of a finite-dimensional linear operator on a special cone. Applying those results to recently introduced tile B-splines, we prove that they can have a higher smoothness than the classical ones of the same order. Moreover, the two-digit tile B-splines have the minimal support of the mask among all refinable functions of the same order of approximation. This proves, in particular, the lowest algorithmic complexity of the corresponding subdivision schemes. Examples and numerical results are provided.

中文翻译：

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各向异性可优化函数和平铺 B 样条曲线


可精炼函数的规律性已在大量文献中进行了分析，并且在两种情况下得到了很好的理解：1） 单变量 2） 具有各向同性膨胀矩阵的多变量。一般（非各向同性）情况提供了很大的阻力。直到 2019 年，非各向同性情况才通过开发矩阵方法完成。在本文中，我们进行下一步，将 Littlewood-Paley 类型方法（在上述特殊情况下非常有效）扩展到具有任意膨胀矩阵的一般方程。这通过特殊圆锥上有限维线性算子的 Perron 特征值给出了 W2k（Rn） 中高阶规则性的公式。将这些结果应用于最近引入的平铺 B 样条曲线，我们证明了它们可以比同阶的经典样条曲线具有更高的平滑度。此外，两位数平铺 B 样条曲线在相同近似阶数的所有可优化函数中具有对掩码的最小支持。这尤其证明了相应细分方案中最低的算法复杂性。提供了示例和数值结果。