Existence of Matrix-Valued Multiresolution Analysis-Based Matrix-Valued Tight Wavelet Frames
Lihong Cui, Ning Zhu, Youquan Wang, Jianjun Sun, Yigang Cen · Numerical Functional Analysis and Optimization · 2016
In this article, we introduce and study the matrix-valued tight wavelet frames for analyzing matrix-valued signal based on matrix-valued multiresolution analysis (MMRA). We put our emphasis on the existence of the MMRA-based matrix-valued tight wavelet frames by establishing the correspondence with their the unitary extension principle (UEP). Here in particular we introduce the square brackets product and the quasi-interpolatory operator, which makes the certificating process for UEP become relatively simple. Some interesting byproducts, such as features on the quasi-interpolatory operator 𝒫n in the matrix-valued function space case, are the critical foundation for our main work.