Multivariate compactly supported C functions by subdivision

This paper discusses the generation of multivariate C degrees degrees 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 + ??????] 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 degrees degrees compactly supported wavelets of small support in any dimension.