Abstract
We consider spaces of bivariate C r -splines of degree k defined on a triangulation of a polygonal domain. The lower bound on the dimension of such spaces given by L. L. Schumaker is valid only for simply connected domains. In this paper, we establish a lower bound on the dimension of such spaces for any polygonal domain possibly with holes. This lower bound is sharp in the sense that it gives the exact dimension for such spaces when k > 3r +1. Furthermore, this result is extended to bivariate spline spaces on arbitrary polygonal partitions.
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