Construction of finite tight frames via singular value decomposition

Abstract

Finite tight frames are used widely for many applications. An important problem is to construct finite frames containing specified elements as needed in real world applications. In this paper we provide a stable method for that purpose. We establish an identity for constructing equi-norm tight frames from the specified elements and present a method for generating equi-norm tight frames explicitly from the outcome of the singular value decomposition of the matrix of the given elements. Under certain conditions, the number of columns of the unit-norm tight frames generated by our method reaches the minimum bound, improving a result in the literature. Finally, we illustrate our method through a numerical example.

Keywords:

• Frames
• tight frame
• tight frame matrix
• singular value decomposition

AMS subject classification:

• 65F30

Acknowledgements

We thank the reviewers for their careful reading of our manuscript and many helpful suggestions, which improve the presentation of the article. The second author participated in the Tenured Faculty Enhancement Program at his institution. This work was made possible in part with the support of the Center for Excellence in Teaching and Learning at Kennesaw State University.

Disclosure statement

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