Construct stable locally supported basis functions

Develop constructive algorithms for stable, locally supported basis functions over general non-diagonalizable T-meshes by using the localized linear equations derived from decoupled vertex cofactors.

Background

The paper derives localized edge equations and a condensed compatibility system for computing spline-space dimensions, but it does not construct basis functions. The authors explicitly leave open the development of constructive algorithms that exploit the decoupled-cofactor formulation to obtain stable, locally supported bases on general non-diagonalizable T-meshes, with motivation from isogeometric analysis.

References

Several promising directions remain open for future research: Extension to Arbitrary and Mixed Order of Smoothness: A natural problem is to extend the decoupling framework to polynomial splines with lower or mixed orders of smoothness ($\mu < d-1$), where the continuity constraints across adjacent cells exhibit more complex algebraic couplings. Basis Function Construction: Utilizing the localized linear equations derived from the decoupled cofactors, we plan to develop constructive algorithms for stable, locally supported basis functions over general non-diagonalizable T-meshes, which is essential for isogeometric analysis (IGA) applications.

— Sharp Dimension Bounds for Spline Spaces over T-meshes with Highest Order of Smoothness  (2608.19839 - Huang et al., 20 Aug 2026) in Section 5, Conclusion and future work