Extension to arbitrary and mixed smoothness orders

Extend the decoupling framework to polynomial spline spaces with lower or mixed orders of smoothness, bc<d-1, in which continuity constraints across adjacent cells exhibit more complex algebraic couplings.

Background

The paper develops a decoupling framework for polynomial splines over T-meshes in the highest-order smoothness case, where the spline degree is d and the smoothness order is d-1. Its local Vandermonde structure and condensed coupling matrix rely on this setting. The authors explicitly identify extending the framework to lower or mixed smoothness orders as an unresolved direction because the associated continuity constraints become algebraically more complicated.

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.

— 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