Regularity of irregular subdivision

We study the smoothness of the limit function for one-dimensional unequally spaced interpolating subdivision schemes. The new grid points introduced a t every level can lie in irregularly spaced locations between old, adjacent grid points and not only midway as is usually the case. For the natural ge neralization of the four-point scheme introduced by Dubuc and Dyn, Levin, a nd Gregory, we show that, under some geometric restrictions, the limit func tion is always C-1; under slightly stronger restrictions we show that the l imit function is almost C-2, the same regularity as in the regularly spaced case.