This talk presents the generation of multivariate C∞ functions with coompact small supports by non-stationary subdivision schemes. Follow- ing the construction of such a univariate function, called ”Up function”, by a non-stationary scheme based on masks of stationary schemes generating B-splines of growing degrees, we term the multivariate functions we generate Up-like functions, and generate them by non-stationary schemes based on masks of stationary schemes generating box-splines of growing supports .

To analyze the convergence and smoothness of these non-stationary schemes, we develped new tools for analyizing convergence and smooth- ness of certain classes of non-stationary schemes which are wider than the class of schemes generating Up-like functions. These new tools are also pre- sented in the talk, as well as a method for achieving small compact supports, by which we obtain in the univariate case Up-like functions with supports [0, 1 + ε], with ε arbitrarily small, in comparison to the support [0, 2] of the Up function.