Matrix splitting with symmetry and dyadic framelet filter banks over algebraic number fields
Tight wavelet frames have many applications, for example, in signal denoising and image processing. One of the most important topics in wavelet analysis is the study of tight wavelet frames and how to construct them with desirable properties. This paper addresses the important problem of how to design tight framelet filter banks with symmetry over algebraic number fields, which are of particular importance in both theory and application. To achieve this goal, the authors investigate the matrix splitting problem, that is, how to split a \(2\times 2\) matrix of Laurent polynomials with symmetry. They obtain a necessary and sufficient condition in Theorem 1 and a step-by-step algorithm in Algorithm 1 on the matrix splitting problem with symmetry. As an application, the authors develop an interesting result in Theorem 2 to construct tight framelet filter banks with symmetry over algebraic number fields. Many convincing and interesting examples are provided in Section 4 to illustrate the results and algorithms developed in this paper.
scientific article; zbMATH DE number 6104564
scientific article