Optimal dual frames for uniform frames with erasures

Given a frame as a coder, its optimal dual as encoder can make the reconstruction error arising from erasure minimum. According to the eigenvalues of the frame operator, we give some sufficient conditions under which the unique optimal dual frame can be identified. Firstly for a Unit-form frame, if the eigenvalues of frame operator |λi| equal to a constant, the canonical dual is the only optimal dual. Secondly, for a uniform frame, if the eigenvalues of frame operator |λi| equal to a constant, the canonical dual is the unique optimal dual. Finally, when we consider the probability of data loss, we will find that if the eigenvalues of frame operator |λi| equal to a constant, the canonical dual is the only probability optimal dual. As an application, we give the examples to show the effectiveness of the theorems we proposed.