Bi-cubic analysis-suitable++ T-splines (AS++ T-splines) (Li in Comput Methods Appl Mech Eng 333:462–474, 2018) are T-splines defined on less restricted T-meshes than analysis-suitable T-splines (AS T-splines), which are both important tools in isogeometric analysis (IGA). In this paper, we generalize the bi-cubic AS++ T-splines to arbitrary degrees and describe some important mathematical properties. Specifically, we develop the conditions under which an AS++ T-spline space belongs to another AS++ T-spline space. This result provides the foundation for the optimized local refinement (Zhang in Comput Methods Appl Mech Eng 342:32–45, 2018) and also is one of the keys for AS++ T-spline approximation. In the end, the optimal approximation properties of the associated T-spline space are developed for arbitrary AS++ T-spline space with the assumption of existence of dual basis, which is automatically satisfied for bi-cubic AS++ T-spline spaces.

Bi-cubic analysis-suitable++ T-splines (AS++ T-splines) (Li in Comput Methods Appl Mech Eng 333:462–474, 2018) 是在比适合分析的 T-splines 限制更少的 T-mesh 上定义的 T-splines ( AS T 样条），它们都是等几何分析 (IGA) 中的重要工具。在本文中，我们将双三次 AS++ T 样条推广到任意程度，并描述了一些重要的数学特性。具体来说，我们开发了一个 AS++ T-spline 空间属于另一个 AS++ T-spline 空间的条件。该结果为优化局部细化提供了基础（Zhang in Comput Methods Appl Mech Eng 342:32–45, 2018），也是 AS++ T-spline 逼近的关键之一。到底，