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 done only recently 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 $W_2^k(\mathbb{R}^n)$ 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.

翻译：可细化函数的正则性已在大量文献中得到分析，并在两种情况下被充分理解：1）单变量情况；2）具有各向同性膨胀矩阵的多变量情况。一般（非各向同性）情况曾面临巨大阻力，直到近期才通过矩阵方法得以解决。本文迈出下一步，将在前述特殊情形下非常有效的Littlewood-Paley型方法推广至具有任意膨胀矩阵的一般方程。这通过特殊锥上的有限维线性算子的Perron特征值，给出了$W_2^k(\mathbb{R}^n)$中高阶正则性的计算公式。将这些结果应用于近期引入的瓦片B样条，我们证明其能达到比同阶经典B样条更高的光滑度。此外，二位数瓦片B样条在所有同阶逼近阶的可细化函数中具有最小支撑掩模。这特别证明了相应细分算法具有最低的算法复杂度。文中给出了实例与数值结果。