On vector-valued nonuniform multiresolution analysis

Abstract

In a vector-valued nonuniform multiresolution analysis (VNUMRA, in short), the corresponding translation set may not be a subgroup of $\mathbb{R}$, but a spectrum associated with one-dimensional spectral pair. In this work, first, we give necessary and sufficient conditions for a family of functions related to VNUMRA to be an orthonormal system and complete system. We provide algebraic results related to the construction of associated vector-valued wavelets from the given VNUMRA. Finally, we give a sufficient condition for the construction of vector-valued wavelets from a given VNUMRA.

Keywords:

• Nonuniform multiresolution analysis
• scaling function
• spectral pair
• vector-valued wavelets
• matrix-valued multiresolution analysis

2010 Mathematics Subject Classifications:

• 42C40
• 42C15
• 42C30
• 42C05

Acknowledgements

The authors are grateful to an anonymous referee for careful reading of the paper and valuable suggestions that have improved the final version of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of the first author is supported by the University Grants Commission (UGC), India [grant number 19/06/2016(i)EU-V]. The second author is partially supported by the Faculty Research Programme Grant-IoE, University of Delhi, Delhi-110007, India [grant number IoE/FRP/PCMS/2020/27].