The Initial Functions in a Subdivision Scheme

In 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 uniied 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 deenition of convergence of subdivision schemes in Banach spaces is also introduced. x1. Introduction Reenable functions play an important role in wavelet analysis. As widely used in computer aided geometric design, a subdivision scheme is a powerful tool in investigating various properties of reenable functions. An s s integer matrix M is called a dilation matrix if lim k!1 M ?k = 0. We say that a is a mask on ZZ s if a is a nitely supported sequence on ZZ s such that P 2ZZ s a() = 1. Wavelets are derived from reenable functions via a standard multiresolution technique. A reenable function is a solution to the following reenement equation = j det Mj X 2ZZ s a()(M ?); (1) where a is a mask and M is a dilation matrix. For a mask a on ZZ s and an s s dilation matrix M, it is known that there exists a unique compactly supported distributional solution, denoted by M a throughout the paper, to the reenement equation (1) such that ^ M a (0) = 1, where the Fourier transform of f 2 L 1 (IR s) is deened to be ^