This paper discusses the generation of multivariate $C^{\infty }$ functions with compact small supports by subdivision schemes. Following the construction of such a univariate function, called Up-function, by a non-stationary scheme based on masks of spline subdivision schemes of growing degrees, we term the multivariate functions we generate Up-like functions. We generate them by non-stationary schemes based on masks of three-directional box-splines of growing supports. To analyze the convergence and smoothness of these non-stationary schemes, we develop new tools which apply to a wider class of schemes than the class we study. With our method for achieving small compact supports, we obtain in the univariate case, Up-like functions with supports $[0,1+ϵ]$ in comparison to the support $[0,2]$ of the Up-function. Examples of univariate and bivariate Up-like functions are given. As in the univariate case, the construction of Up-like functions can motivate the generation of $C^{\infty }$ compactly supported wavelets of small support in any dimension.

本文讨论了多元变量的生成$C^{无穷大}$通过细分方案提供紧凑的小型支撑功能。在通过基于增长程度的样条细分方案掩码的非平稳方案构造出这样的单变量函数（称为Up 函数）之后，我们将生成的多元函数称为类 Up 函数。我们通过基于增长支撑的三向箱形样条掩模的非平稳方案来生成它们。为了分析这些非平稳方案的收敛性和平滑性，我们开发了新的工具，这些工具适用于比我们研究的类别更广泛的方案。通过我们实现小型紧凑支撑的方法，我们在单变量情况下获得了具有支撑的类似 Up 的函数$[0,1+\epsilon ]$与支持相比$[0,2]$的上功能。给出了单变量和双变量 Up-like 函数的示例。与单变量情况一样，类似 Up 的函数的构造可以激发$C^{无穷大}$任意维小支撑的紧支撑小波。