Scalar and Hermite subdivision schemes (Q861339)

A criterion of convergence for stationary nonuniform subdivision sche\-mes is provided. For periodic subdivision schemes, this criterion is optimal and can be applied to Hermite subdivision schemes which are not necessarily interpolatory. For the Merrien family of Hermite subdivision schemes which involve two parameters, the values of the parameters for which the Hermite subdivision scheme is convergent can be described explicitly. The paper is organized as follows. The basic terminology for subdivision schemes is described in Section 2. In Section 3, the authors obtain a difference subdivision scheme from an affine subdivision scheme. Section 4 contains a criterion of convergence for nonuniform subdivision schemes. In Section 5, the authors study the basic properties of \(C^1\)-convergence and the reproduction of linear functions for Hermite subdivision schemes. In Section 6, they associate a nonuniform scalar subdivision scheme to any interpolating Hermite subdivision scheme that reproduces constants, and obtain a necessary and sufficient condition for \(C^1\)-convergence for a given Hermite subdivision scheme that reproduces linear functions. Section 7 recalls the Merrien family of Hermite subdivision schemes. Section 8 contains necessary conditions for convergence in the Merrien family. These conditions are in fact sufficient, and in Section 9, authors obtain the precise definition of the convergence region, and provide examples in Section~10.