A Preliminary Study on the Dimensional Stability Classification of Polynomial Spline Spaces over T-meshes

This paper introduces the concept of dimensional stability for spline spaces over T-meshes, providing the first mathematical definition and a preliminary classification framework. We define dimensional stability as an invariant within the structurally isomorphic class, contingent on the rank stability of the conformality matrix. Absolute stability is proposed via structurally similar maps to address topological and order structures. Through the $k$-partition decomposition of T-connected components and analysis of the CNDC, we establish a correspondence between conformality vector spaces and rank stability. For diagonalizable T-meshes, decomposition into independent one-dimensional T $l$-edges facilitates basis function construction, while arbitrary T-meshes are partitioned into one- and two-dimensional components. These findings lay the groundwork for understanding dimensional stability and developing spline space basis functions.