On approximation properties of matrix-valued multi-resolution analyses

Abstract

We study approximation properties of multi-resolution analyses in the context of matrix-valued function spaces. Here, we generalize the notions of approximation order and density order given by the reference [de Boor C, DeVore RA, Ron A. Approximation from shift-invariant subspaces of $L^2({\mathbb{R}}^d)$. Trans Am Math Soc. 1994;341(2):787–806]. Indeed, we prove a characterization of the closed subspaces generated by the shifts of a single matrix-valued function that provide approximation order and/or density order $\alpha \ge 0$. To give our conditions, we need the classical notion of approximate continuity. As a consequence, we prove necessary and sufficient conditions on a matrix-valued function to be a scaling function in a multi-resolution analysis.

Keywords:

• Approximate continuity
• approximation and density order
• Fourier transform
• matrix-valued multi-resolution analysis
• scaling functions

2000 Mathematics Subject Classifications:

• 42C40
• 41A30

Acknowledgments

We would like to thank the anonymous referees for their valuable comments. In particular, for their suggestions to clarify the proof of the necessary condition in Theorem 4.1.

Disclosure statement

No potential conflict of interest was reported by the authors.