Subdivision and spline spaces
This work provides theoretical foundations for spline spaces under mesh refinement, benefiting researchers in approximation theory and geometric modeling.
The paper studies mesh refinement in approximation theory, providing sufficient conditions for the spline module on a subdivided mesh to decompose as a direct sum of splines on the original mesh and the subdivided cell, leading to dimension formulas and explicit bases for common subdivisions.
A standard construction in approximation theory is mesh refinement. For a simplicial or polyhedral mesh D in R^k, we study the subdivision D' obtained by subdividing a maximal cell of D. We give sufficient conditions for the module of splines on D' to split as the direct sum of splines on D and splines on the subdivided cell. As a consequence, we obtain dimension formulas and explicit bases for several commonly used subdivisions and their multivariate generalizations.