Convergence of subdivision schemes in L p spaces

In this paper it is proved that L _p solutions of a refinement equation exist if and only if the corresponding subdivision scheme with suitable initial function converges in L _p without any assumption on the stability of the solutions of the refinement equation. A characterization for convergence of subdivision scheme is also given in terms of the refinement mask. Thus a complete answer to the relation between the existence of L _p solutions of the refinement equation and the convergence of the corresponding subdivision schemes is given. Supported by the National Natural Science Foundation of China(10071071).