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

TL;DR

This paper defines and classifies the concept of dimensional stability in polynomial spline spaces over T-meshes, establishing foundational theoretical results and decomposition methods for basis function construction.

Contribution

It introduces the first mathematical definition of dimensional stability and provides a classification framework, including decomposition techniques for T-meshes and analysis of conformality matrices.

Findings

Defined dimensional stability as an invariant related to conformality matrix rank stability.

Established a correspondence between conformality vector spaces and rank stability.

Developed decomposition methods for T-meshes to facilitate basis function construction.

Abstract

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.