Matrix methods for perfect signal recovery underlying range space of operators
• Fahimeh Arabyani Neyshaburi,
• Ramin Farshchian,
• Rajab Ali Kamyabi-Gol
Fahimeh Arabyani Neyshaburi

Corresponding Author:arabianif@yahoo.com

Author Profile
Ramin Farshchian
Author Profile
Rajab Ali Kamyabi-Gol
Author Profile

## Abstract

The purpose of this work is to investigate perfect reconstruction underlying range space of operators in finite dimensional Hilbert spaces by a new matrix method. To this end, first we obtain more structures of the canonical $K$-dual. % and survey optimal $K$-dual problem under probabilistic erasures. Then, we survey the problem of recovering and robustness of signals when the erasure set satisfies the minimal redundancy condition or the $K$-frame is maximal robust. Furthermore, we show that the error rate is reduced under erasures if the $K$-frame is of uniform excess. Toward the protection of encoding frame (K-dual) against erasures, we introduce a new concept so called $(r,k)$-matrix to recover lost data and solve the perfect recovery problem via matrix equations. Moreover, we discuss the existence of such matrices by using minimal redundancy condition on decoding frames for operators. We exhibit several examples that illustrate the advantage of using the new matrix method with respect to the previous approaches in existence construction. And finally, we provide the numerical results to confirm the main results in the case noise-free and test sensitivity of the method with respect to noise.
28 Nov 2021Submitted to Mathematical Methods in the Applied Sciences
08 Dec 2021Submission Checks Completed
08 Dec 2021Assigned to Editor
10 Dec 2021Reviewer(s) Assigned
18 Aug 2022Review(s) Completed, Editorial Evaluation Pending
18 Aug 2022Editorial Decision: Revise Minor