Bivariate splines on Tohǎneanu partition

Abstract In Tohǎneanu (2005), Tohǎneanu introduced a triangulation Δ T with the following property: the space of splines of smoothness r and degree 2 r defined on Δ T has a dimension not equal to the lower bound found by Schumaker in Schumaker (1979). In Minac and Tohǎneanu (2013), it was shown that for d ≥ 2 r + 1 , the dimension of the space of splines on Δ T is equal to the lower bound. We show that this phenomenon can be explained by intrinsic supersmoothness of splines. Moreover, using this supersmoothness we obtain the dimension of the space of splines of smoothness r and degree ≤ 2 r on Δ T .