Convergence of cascade algorithms and smoothness of refinable distributions (Q1431019)

Let \(F\) be an \(N\times 1\) vector of functions in a function space \(X\). In this paper, the author considers the cascade algorithm \(\{ T^n F\}_{n=1}^\infty\), where the cascade operator \(T\) on \(X^N\) is defined to be \[ TF:=\sum_{j\in \mathbb{Z}^d} c(j) F(2\cdot-j), \] and \(c(j), j\in \mathbb{Z}^d\) are \(N\times N\) matrices and \(c(j)=0\) for all but finitely many \(j\in \mathbb{Z}^d\). The \(\{c(j)\}_{j\in \mathbb{Z}^d}\) is called the mask. Let \(\mathcal{D}_r\) denote the space of all compactly supported \(C^\infty\) functions \(h\) on \(\mathbb{R}^d\) such that \(\hat h(\xi)=O(| \xi| ^r)\) as \(\xi\to 0\). By introducing an interesting notion of the dependent ideal \(i_r(F):=\{ (h*F(j))_{j\in\mathbb{Z}^d} : h\in \mathcal{D}_r\}\) of \(F\), the author is able to obtain in Theorem~2.1 the asymptotic behavior of the cascade sequence \(T^n F\) in a family of function spaces without the assumption of stability on the initial function vector \(F\). As a consequence, the author characterizes the existence of a refinable function vector \(\Psi\) (that is, \(T \Psi=\Psi\)) in \(L^p\) and establishes the convergence of the cascade algorithm in various function spaces. A criterion for a refinable function vector \(\Psi\in L^{p, \gamma}\) has been given in Theorem~3.1. The author also discusses (mainly for the scalar case \(N=1\)) in Section 4 how to choose an appropriate initial function vector in a cascade algorithm.