Refinable functions from their values at integers (Q1399082)

Let \(\phi^0\) be a continuous function on \(\mathbb{R}\) such that \(\phi^0\) is positive on \((0, N+1)\) for some positive integer \(N\) and zero otherwise, which satisfies the equation \(\sum_{j\in \mathbb{Z}} \phi^0(\cdot-j)=1\). For \(\theta\in (0,1)\), define \[ a^0(z):=\sum_{j\in \mathbb{Z}} L_j(\phi^0)z^j, \quad z\in \mathbb{C}, \] where \(L_j(\phi^0)=\theta \phi^0(j-1)+\phi^0(j)+(1-\theta)\phi^0(j+1)\), \(j\in \mathbb{Z}\). Then there is a sequence of continuous refinable functions \(\{ \phi^n\, : \, n\in \mathbb{Z}_+\}\) such that \(\sum_{j\in \mathbb{Z}} \phi^{n+1}(\cdot-j)=1\) and \(\phi^{n+1}=\sum_{j\in \mathbb{Z}} L_j(\phi^n)\phi^{n+1}(2\cdot-j)\). In this paper, the authors investigate some interesting properties such as convergence and rate of convergence of this sequence of refinable functions \(\{\phi^n\}_{n\in \mathbb{Z}_+}\). Let \(\phi^\infty\) denote the limit function of the sequence \(\{\phi^n\}_{\mathbb{Z}_+}\). The authors obtain a sharp error estimate of \(\| \phi^n-\phi^\infty\| _{C(\mathbb{R})}\). Perturbation of refinable functions has also been discussed in this paper.