On Dimension Extension of a Class of Iterative Equations

Abstract

This investigation aims at studying some special properties (convergence, polynomial preservation order, and orthogonal symmetry) of a class of r-dimension iterative equations, whose state variables are described by the following nonlinear iterative equation: ${\varphi }^n(x)=T({\varphi }^{n-1}(x)):={\sum }_{j=0}^m{H}_j{\varphi }^{n-1}(2x-k)$. The obtained results in this paper are complementary to some published results. As an application, we construct orthogonal symmetric multiwavelet with additional vanishing moments. Two examples are also arranged to demonstrate the correctness and effectiveness of the main results.