We analyze the space of bivariate functions that are piecewise polynomial ofbi-degree \textless{}= (m, m') and of smoothness r along the interior edges ofa planar T-mesh. We give new combinatorial lower and upper bounds for thedimension of this space by exploiting homological techniques. We relate thisdimension to the weight of the maximal interior segments of the T-mesh, definedfor an ordering of these maximal interior segments. We show that the lower andupper bounds coincide, for high enough degrees or for hierarchical T-mesheswhich are enough regular. We give a rule of subdivision to constructhierarchical T-meshes for which these lower and upper bounds coincide. Finally,we illustrate these results by analyzing spline spaces of small degrees andsmoothness.
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