Bases of T-meshes and the refinement of hierarchical B-splines

In this paper we consider spaces of bivariate splines of bi–degree (m,n) with maximal order of smoothness over domains associated to a two–dimensional grid. We define admissible classes of domains for which suitable combinatorial technique allows us to obtain the dimension of such spline spaces and the number of tensor–product B–splines acting effectively on these domains. Following the strategy introduced recently by Giannelli and Jüttler, these results enable us to prove that under certain assumptions about the configuration of a hierarchical T–mesh the hierarchical B-splines form a basis of bivariate splines of bi–degree (m,n) with maximal order of smoothness over this hierarchical T–mesh. In addition, we derive a sufficient condition about the configuration of a hierarchical T-mesh that ensures a weighted partition of unity property for hierarchical B-splines with only positive weights.