Dimensions and bases of hierarchical tensor-product splines

Abstract : We prove that the dimension of trivariate tensor-product spline space of tri-degree (m,m,m) with maximal order of smoothness over a three- dimensional domain coincides with the number of tensor-product B-spline basis functions acting effectively on the domain considered. A domain is required to belong to a certain class. This enables us to show that, for a cer- tain assumption about the configuration of a hierarchical mesh, hierarchical B-splines span the spline space. This paper presents an extension to three-dimensional hierarchical meshes of results proposed recently by Giannelli and Ju ̈ttler for two-dimensional hierarchical meshes.
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Dmitry Berdinsky, Taiwan Kim, Cesare Bracco, Durkbin Cho, Bernard Mourrain, et al.. Dimensions and bases of hierarchical tensor-product splines. Journal of Computational and Applied Mathematics, Elsevier, 2014, 257, pp.86-104. ⟨10.1016/j.cam.2013.08.019⟩. ⟨hal-00876557⟩

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