Multiple refinable Hermite interpolants
We consider solutions of a system of refinement equations written in the form as \(\varphi(x)= \sum_{n\in \mathbb{Z}}a(n) \varphi(2x-n)\), where \(\varphi= (\varphi_1, \dots, \varphi_r)^T\) is a vector of compactly supported functions on \(\mathbb{R}\) and \(a\) is a finitely supported sequence of \(r\times r\) matrices called the refinement mask. If \(\varphi\) is a continuous solution and \(a\) is supported on \([N_1,N_2]\), then \(v:=(\varphi (n))^{N_2-1}_{n=N_1}\) is an eigenvector of the matrix \((a(2k-n))^{N_2-1}_{k,n=N_1}\) associated with eigenvalue 1. Conversely, given such an eigenvector \(v\), we may ask whether there exists a continuous solution \(\varphi\) such that \(\varphi(n)= v(n)\) for \(N_1\leq n\leq N_2-1\) \((\varphi(n)=0\) for \(n\notin [N_1,N_2-1]\), according to the support). The first part of this paper answers this question completely. This existence problem is more general than either the convergence of the subdivision scheme or the requirement of stability, since in one of the latter cases, the eigenvector \(v\) is unique to a constant multiplication. The second part of this paper is concerned with Hermite interpolant solutions, i.e., for some \(n_0\in \mathbb{Z}\) and \(j,m=1, \dots,r\), \(\varphi_j\in C^{r-1} (\mathbb{R})\) and \(\varphi_j^{(m-1)}(n)= \delta_{j,m} \delta_{n,n_0}\), \(n\in\mathbb{Z}\). We provide a necessary and sufficient condition for the refinement equation to have an Hermite interpolant solution. The condition is strictly in terms of the refinement mask. Our method is to characterize the existence and the Hermite interpolant condition by joint spectral radii of matrices. Several concrete examples are presented to illustrate the general theory.
scientific article; zbMATH DE number 1417551
scientific article