A Lower Bound on the Dimension of Bicubic Spline Spaces over T-meshes

In this paper, we discusses the dimensions of the bicubic spline spaces over T-meshes. Specially, we use two concepts: extension of T-meshes and spline spaces with homogeneous boundary conditions. In the dimension analysis, the important technique is linear space embedding with the operator of mixed partial derivative, which embeds the space of higher order into the space of lower order. Similar with the discussion of the dimension of biquadratic spline spaces over T-meshes, the necessary and sufficient conditions are described by the operator. Using the characteristic of T-meshes, we can reduce the number of conditions. With this method, a dimension lower bound of bicubic spline spaces over regular T-meshes can be provided. It is only depends on the topology of the T-meshes.