Existence of Matrix-Valued Multiresolution Analysis-Based Matrix-Valued Tight Wavelet Frames

ABSTRACT

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 𝒫_n in the matrix-valued function space case, are the critical foundation for our main work.

KEYWORDS:

• Matrix-valued tight wavelet frames
• unitary extension principle
• MMRA
• quasi-interpolatory operator

MATHEMATICS SUBJECT CLASSIFICATION:

• 42C15
• 42C40

Acknowledgments

The authors would like to thank Professor Jian-ao Lian (Prairie View A&M University) for useful discussions and suggestions by numerous email exchanges. The first author is also very grateful to the Mathematics Department of The Georgia Institute of Technology for providing her a very good office environment.