On the problem of instability in the dimension of a spline space over a T-mesh

In this paper, we discuss the problem of instability in the dimension of a spline space over a T-mesh. For bivariate spline spaces S (5, 5, 3, 3) and S (4, 4, 2, 2), the instability in the dimension is shown over certain types of T-meshes. This result could be considered as an attempt to answer the question of how large the polynomial degree (m,m) should be relative to the smoothness (r, r) to make the dimension of a spline space stable. We show in particular that the bound m ≥ 2r + 1 and m ≥ 2r + 1 are optimal.