Spline space and its B-splines on an n+1 direction mesh in Rn

It is well known that splines play an important role in many fields, especially, their close relationship with wavelets makes them have more widespread applications in numerous scientific and engineering domains. Univariate and bivariate splines have been well studied and lots of results have been obtained. Because of the intrinsic difficulty between bivariate case and higher-dimension (three or more dimensions) settings, the study of splines on higher dimensions are very limited. For example, the study of the bivariate splines on a three-direction mesh triangulation has obtained many important and excellent results, but almost all of those results have no analog generalization to higher dimensions. In this paper, we will study the higher-dimension splines defined on n+1 mesh simplical partitions which is the analog of bivariate splines on three-mesh triangulations. We have also pointed out many interesting differences between bivariate splines and higher-dimensional cases. Our main results are that, similar to bivariate and trivariate cases, a necessary and sufficient condition for Skr(Δ) to contain a B-spline is k⩾12(r+1)(n+1) for r⩾1 being odd and k⩾12r(n+1)+1 for r⩾0 being even.