On the stability of the dimensions of spline spaces with highest order of smoothness over T-meshes

This paper studies the stability of the dimension of the spline space Sd(J) of bi-degree (d, d) with highest order of smoothness over a T-mesh J. By decomposing the T-connected component of J into a diagonalizable component and a non-diagonalizable component, we prove that the stability of the dimension of the spline space Sd(J) depends only on the stability of the rank of a matrix M corresponding to the multi-vertices of the non-diagonalizable component. The matrix M has a much smaller size than the conformality matrix associated with the T-connected component, and thus the stability is much easier to verify. As an application, we reprove the stability of the dimension of the spline space S3(J) over a T-mesh which is generated by subdividing a collection of 2 x 2 submeshes of a tensor product mesh under cross subdivision.