Han, Bin2007-08-162007-08-162001-07https://hdl.handle.net/11299/3657In this paper we shall study the initial functions in a subdivision scheme in a Sobolev space. By investigating the mutual relations among the initial functions in a subdivision scheme, we are able to study in a relatively unified approach several questions related to a subdivision scheme in a Sobolev space such as convergence, error estimate and convergence rate of a subdivision scheme in a Sobolev space with a general dilation matrix. A generalized definition of convergence of subdivision schemes in Banach spaces is also introduced.