Tchebycheffian spline spaces over planar T-meshes: Dimension bounds and dimension instabilities

We consider Tchebycheffian spline spaces over planar T-meshes and we study their dimension. We show that the structure of extended Tchebycheff spaces allows us to fully generalize the dimension upper bounds known in the literature for polynomial spline spaces over T-meshes. Moreover, we illustrate that the dimension of Tchebycheffian spline spaces over T-meshes can be unstable for certain configurations of the T-mesh, for any choice of the underlying extended Tchebycheff space.