In this paper we study the dimension of splines of mixed smoothness on axis-aligned T-meshes. This is the setting when different orders of smoothness are required across the edges of the mesh. Given a spline space whose dimension is independent of its T-mesh's geometric embedding, we present constructive and sufficient conditions that ensure that the smoothness across a subset of the mesh edges can be reduced while maintaining stability of the dimension. The conditions have a simple geometric interpretation. Examples are presented to show the applicability of the results on both hierarchical and non-hierarchical T-meshes. For hierarchical T-meshes it is shown that mixed smoothness spline spaces that contain the space of PHT-splines (Deng et al., 2008) always have stable dimension.

在本文中，我们研究了在轴对齐的T形网格上混合平滑度的样条曲线的维数。当在网格的边缘上需要不同顺序的平滑度时，此设置。给定一个样条空间，其尺寸与其T网格的几何嵌入无关，我们提出了建设性的充分条件，可确保在保持尺寸稳定性的同时，减小网格边缘的子集的平滑度。条件具有简单的几何解释。给出了示例以显示结果在分层和非分层T-网格上的适用性。对于分层的T网格，表明包含PHT样条空间的混合平滑样条空间（Deng等，2008）始终具有稳定的尺寸。