Subdivision and Spline Spaces

A standard construction in approximation theory is mesh refinement. For a simplicial or polyhedral mesh \(\varDelta \subseteq {\mathbb R}^k\), we study the subdivision \(\varDelta '\) obtained by subdividing a maximal cell of \(\varDelta \). We give sufficient conditions for the module of splines on \(\varDelta '\) to split as the direct sum of splines on \(\varDelta \) 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.