Frame properties of generalized shift-invariant systems in discrete setting

Abstract

In this paper, we present a simple characterization of those sequences for which purely shift-invariant systems are normalized tight frames for . As an application, a characterization for the Gabor system to be a normalized tight frame can be directly derived. Further, we prove that if the Calderón condition holds, then such purely shift-invariant systems are still normalized tight frames for . At the end of the paper, a sufficient condition for those purely shift-invariant systems to be frames for is established which is a variant of the classical wavelet systems to be frames for .

Keywords:

• Frame
• shift-invariant system
• affine system
• Gabor system
• wavelet
• Calderón condition

AMS Subject Classifications:

• 42C15
• 42C40

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 61471410], the Natural Science Foundation for the Education Department of Henan Province of China [grant number 13A110072] and the Foundation for Science and Technology Department of Henan Province of China [grant number 132300410150].