α-Phase retrieval frame in Hilbert space and its application

In the present work, the properties of α-phase retrieval frame with a redundant set of vectors under perturbation of the frame set have been discussed. We provided the necessary and sufficient condition to exist in the exact reconstruction using α-phase retrieval frame. The deletion of the noise for the reconstructed signal has been discussed. Finally, this paper provided if bounded nonlinear operator Lα on infinite dimensional complex Hilbert space is a global phase shift, then Kernel of the operator is spanning a set of the vector.