Tight frames, partial isometries, and signal reconstruction

Abstract

This article gives a procedure to convert a frame which is not a tight frame into a Parseval frame for the same space, with the requirement that each element in the resulting Parseval frame can be explicitly written as a linear combination of the elements in the original frame. Several examples are considered, such as a Fourier frame on a spiral. The procedure can be applied to the construction of Parseval frames for the space of square integrable functions whose domain is the ball of radius When a finite number of measurements is used to reconstruct a signal in error estimates arising from such approximation are discussed.

Keywords:

• Fourier frames
• tight frames
• partial isometry
• polar decomposition

AMS Subject Classifications:

• 42C15
• 47A50

Acknowledgments

The authors are immensely grateful to John Benedetto for being a constant source of inspiration and a mathematical role model. The second named author also wishes to express sincere gratitude to Doug Cochran and John McDonald for useful discussions on the topic. Finally, the authors thank the anonymous reviewers for their suggestions.

Notes

1 The first named author is supported by a postdoctoral fellowship from the Pacific Institute for the Mathematical Sciences. The second named author was partially supported by AFOSR [grant number FA9550-10-1-0441].