Convergence analysis of $C^2$ Hermite interpolatory subdivision schemes by explicit joint spectral radius formulas

Subdivision schemes are popular iterative processes to build graphs of functions, curves and surfaces. We analyze the 2-point Hermite C2 subdivision scheme introduced by Merrien in [26]. For the analysis of its convergence and its smoothness properties we are concerned with the computation of the joint spectral radius of a family of 2 matrices associated with the scheme. In this paper, by an explicit computation of the joint spectral radius of such pairs of matrices, we determine necessary and sufficient conditions for the scheme to be C2 convergent, whenever it reproduces cubic polynomials. In addition, we present two one-parameter families of convergent subdivision schemes belonging to the class in [26] possessing interesting properties from the shape control point of view.