Averaged numerically optimal dual frames for erasures

The problem of erasures can occur during the transmission of data or signals. Finding the optimal dual frames which can minimize the reconstruction error when erasures occur is a deep problem in frame theory. In this paper, we introduce a new measurement for the error operator by taking the average of the operator norm and the numerical radius. Optimal dual frames obtained using this new measurement are called averaged numerically optimal dual frames, in short, ANOD-frames. We study the properties of ANOD-frames. We prove that the set of all averaged numerically optimal dual frames is convex and compact. It is shown that the images of an ANOD-frame under a unitary operator is also an ANOD-frame. We give some equivalent and sufficient conditions for canonical dual to be the unique averaged numerically optimal dual frame or a nonunique averaged numerically optimal dual frame. In addition, we prove that an averaged numerically optimal canonical dual of a Parseval frame for r-erasures is also an optimal dual frame for r-erasures.