Multivariate refinement equations and convergence of cascade algorithms in \(L_p(0<p<1)\) spaces
We consider the solutions of refinement equations written in the form \[ \varphi(x)= \sum_{\alpha\in \mathbb{Z}^s} a(\alpha) \varphi(Mx- \alpha)+ g(x),\qquad x\in \mathbb{R}^s, \] where the vector of functions \(\varphi=(\varphi_1,\dots, \varphi_r)^T\) is unknown, \(g\) is a given vector of compactly supported functions on \(\mathbb{R}^s\), \(a\) is a finitely supported sequence of \(r\times r\) matrices called the refinement mask, and \(M\) is an \(s\times s\) dilation matrix with \(m= |\text{det }M|\). Inhomogeneous refinement equations appear in the constructions of multiwavelets and the constructions of wavelets on a finite interval. The cascade algorithm with mask \(a\), \(g\), and dilation \(M\) generates a sequence \(\varphi_n\), \(n= 1,2,\dots\), by the iterative process \[ \varphi_n(x)= \sum_{\alpha\in \mathbb{Z}^s} a(\alpha) \varphi_{n-1}(Mx- \alpha)+ g(x),\qquad x\in\mathbb{R}^s, \] from a starting vector of the function \(\varphi_0\). We characterize the \(L_p\)-convergence \((0<p<1)\) of the cascade algorithm in terms of the \(p\)-norm joint spectral radius of a collection of linear operators associated with the refinement mask. We also obtain a smoothness property of the solutions of the refinement equations associated with the homogeneous refinement equation.
scientific article; zbMATH DE number 1921181
scientific article