We consider the univariate two-scale refinement equation.The paper analyzes the correlation between the existence of smoothcompactly supported solutions of this equation and the convergence ofthe corresponding cascade algorithm/subdivision scheme. We introducea criterion that expresses this correlation in terms of mask of theequation. We show that the convergence of subdivision scheme dependson values that the mask takes at the points of its generalizedcycles. This means in particular that the stability of shifts ofrefinable function is not necessary for the convergence of thesubdivision process. This also leads to some results on the degree ofconvergence of subdivision processes and on factorizations ofrefinable functions.
refinement equations cascade algorithm subdivision process degree of convergence stability cycles tree