Abstract
This investigation aims at studying some special properties (convergence, polynomial preservation order, and orthogonal symmetry) of a class ofr-dimension iterative equations, whose state variables are described by the following nonlinear iterative equation:ϕn(x)=T(ϕn−1(x)):=∑j=0mHjϕn−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.
Highlights
IntroductionGiving any compact supported vector-valued function φ0(x) := (φ10, . . . , φr0)⊤ ∈ (L2(R))r, we define r-dimension iterative equation as follows:
Giving any compact supported vector-valued function φ0(x) := (φ10, . . . , φr0)⊤ ∈ (L2(R))r, we define r-dimension iterative equation as follows:μ φn (x) = T (φn−1 (x)) := ∑Hjφn−1 (2x − j), (1)j=0 where Hj is r-order real matrix, j = 0, 1, . . . , μ, μ ∈ Z+, and n ∈ Z+
Using Fourier transform, from (1), (4), and (5), we obtain that the iterative equation (1) converges to vector function φ(x) if and only if the infinite product (5) converges
Summary
Giving any compact supported vector-valued function φ0(x) := (φ10, . . . , φr0)⊤ ∈ (L2(R))r, we define r-dimension iterative equation as follows:. In order to construct multiwavelets with vanishing moments of arbitrarily high order, Abstract and Applied Analysis in [4], with the help of dimension extension and iterative scheme for revising masks of (1), Chui and Lian investigate the compactly supported orthogonal scaling function with additional polynomial preservation order (p.p.o.). The orthogonal symmetric scaling vector function with high p.p.o. is very important to construct symmetric multiwavelets with high vanishing moments by multiresolution analysis, and yet its construction is very difficult, especially in high dimension. The main objective of this paper is to develop an iterative equation to generate orthogonal symmetric scaling vector function as the limit of (1) and to analysis its convergence. We will construct compactly supported orthogonal symmetric multiwavelets to achieve any order vanishing moments
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