  This paper discusses the dimensions of the spline spaces over T-meshes with
lower degree. Two new concepts are proposed: extension of T-meshes and spline
spaces with homogeneous boundary conditions. In the dimension analysis, the key
strategy is linear space embedding with the operator of mixed partial
derivative. The dimension of the original space equals the difference between
the dimension of the image space and the rank of the constraints which ensuring
any element in the image space has a preimage in the original space. Then the
dimension formula and basis function construction of bilinear spline spaces of
smoothness order zero over T-meshes are discussed in detail, and a dimension
lower bound of biquadratic spline spaces over general T-meshes is provided.
Furthermore, using level structure of hierarchical T-meshes, a dimension
formula of biquadratic spline space over hierarchical T-meshes are proved. A
topological explantation of the dimension formula is shown as well.
