Convergence of cascade algorithms in Sobolev spaces for perturbed refinement masks
Let \(\Omega\) be a finite set in \(\mathbb{Z}^s\) and \(M\) be an \(s\times s\) dilation matrix. Let \(a\) be a scalar mask supported on \(\Omega\) such that the cascade algorithm associated with mask \(a\) and dilation \(M\) converges in the Sobolev space \(W_p^n(\mathbb{R}^s)\); that is, \(Q_a^k \phi_0, k\in \mathbb{N}\), is a Cauchy sequence in \(W_p^n(\mathbb{R}^s)\) for all appropriate initial functions \(\phi_0\), where the cascade operator \(Q_a\) is defined to be \[ Q_a f:=\sum_{\beta\in \mathbb{Z}^s} a(\beta) f(M\cdot-\beta). \] It has shown in this paper that there exists \(\eta>0\) such that for any mask \(b\) which is supported on \(\Omega\) with \(\|a-b\|_1< \eta\) and \(b\) satisfies the sum rules of order \(n+1\), then the cascade algorithm associated with mask \(b\) and dilation \(M\) also converges in \(W_p^n(\mathbb{R}^s)\). Moreover, for the initial function \(\phi_{0,n}=N_{(n+1, \ldots, n+1)}\) which is a tensor product \(B\)-spline function of order \(n+1\), there exists a positive constant \(C\), which is independent of \(b\) and \(n\), such that \[ \|Q_a^k \phi_{0,n}-Q_b^k \phi_{0,n}\|_{W_p^n(\mathbb{R}^s)}\leq C\|a-b\|_1 \qquad \forall k\in \mathbb{N} \] and consequently, \[ \|\phi_a-\phi_b\|_{W_p^n(\mathbb{R}^s)}\leq C\|a-b\|_1, \] where \(\phi_a\) and \(\phi_b\) are the \(M\)-refinable functions such that \(Q_a\phi_a=\phi_a\), \(Q_b\phi_b=\phi_b\) and \(\hat \phi_a(0)=\hat \phi_b(0)=1\). When \(n=0\), the above sharp error estimate has been first established by the reviewer in an earlier work and in his PhD thesis. Removing the restriction on the initial function in the above result, later on the reviewer has further shown that the above result holds for a general initial function and a similar nontrivially generalized result also holds for a vector cascade algorithm with a matrix mask. In order to establish the above result, the authors used \(\ell_p\)-norm joint spectral radius and inequalities related to the differentiation of functions in a Sobolev space have been established.
scientific article; zbMATH DE number 1891510